Extended Graph Rotation Systems as a Model for Cyclic Weaving on Orientable Surfaces

Transcription

1 Extended Graph Rotation Systems as a Model for Cyclic Weaving on Orientable Surfaces ERGUN AKLEMAN, JIANER CHEN Texas A&M University, College Station, Texas JONATHAN L. GROSS Columbia University, Ne York Abstract We present an extension of the theory of graph rotation systems, hich has been a idely used model for graph imbeddings on topological surfaces. The extended model is quite beyond hat is needed to specify graph imbeddings on surfaces, and it can be used to represent and generate link structures immersed on surfaces. Since link structures immersed on surfaces can be vieed as oven images in 3D space, the extended graph rotation systems provide a ell-formulated mathematical model for developing a topologically robust graphics system ith strong interactive operations for the design of oven images in 3D mesh-modeling and computer-aided sculpting. 1 Introduction Some of our recent research [AkCh99, AkChSr03, AkChGr09] in computer graphics reveals ho classical topological graph theory [GrTu87, MoTh01, Wh01], especially the theory of graph rotation systems, can be used as a mathematical foundation in the development of a general paradigm for 3D mesh-modeling systems and computer-aided sculpting. In the current paper, e propose a concept of extended graph rotation systems (abbr. EGRS) and study its mathematical properties. Figure 1 shos some examples of the images of cyclic plain-oven objects that ere designed and generated using our graphics softare developed based on the extended graph rotation systems. Venus Bunny Rocker Arm Genus-3 Object Figure 1: Examples of oven objects constructed from orientable manifold meshes. 1

2 Extended Graph Rotation Systems as a Model for Cyclic Weaving 2 Extended graph rotation systems are a generalized version of the standard graph rotation systems, ith a generalized edge-tist operation. Most importantly, instead of regarding a graph rotation system as a combinatorial representation of the graph imbedded on a 2-dimensional manifold (i.e., a surface), e interpret the face boundary alks of the imbedding induced by an extended graph rotation system as a topological link (i.e., a collection of inter-linked knots [Ad04]) immersed on a surface. We sho in this paper ho the extended graph rotation systems are conveniently and effectively used for generating, representing, and operating on link structures immersed on surfaces. When vieing link structures immersed on surfaces as oven structures in 3D-space, the theory of extended graph rotation systems provides a solid mathematical foundation, and it offers poerful techniques for the development of an interactive-graphics system for the design of a variety of 3D oven images. Adhering to the formal mathematical model presented herein assures topological robustness, hich means that the objects generated by our system are alays valid link immersions on surfaces, and completeness, hich means that it can generate link immersions on all possible surfaces. From the perspective of graphic artists, the EGRS model underlying the 3D-modeling softare system provides a systematic method for dealing ith 3D-shapes of complicated topological structures, and it overcomes the problem of large gaps in the eaving structures on surfaces of shapes like those shon in Figure 1. The level of careful detail in Sections 2, 3, and 4 is intended to make implementation of a computer graphics system based on rotation systems reasonably accessible to those beyond the topological graph theory community. In particular, Section 2 provides the prerequisite general background in topological graph theory. As edgetisting, edge-insertion, and edge-deletion are fundamental surgery operations in graph imbeddings, and are critical to our study on the EGRS model, e re-examine these operations in Sections 3 and 4. More specifically, for each of the operations, e identify certain subtle cases that seem not to have been explicated heretofore in the literature of topological graph theory. We propose algorithms to handle these subtle cases and prove the correctness of our algorithms (see Theorem 3.4, Case (B2), Theorem 4.1, Case (B2), and Theorem 4.2, Case (B2)). Of course, e cannot provide a mathematical guide to the artistic inspiration involved in the creation of the oven objects in Figure 1. Hoever, from the perspective of a topologist, a graphics softare system based on the EGRS model, such as TOPMOD [ACSMT08], enables an artist 1. to imbed a 1-skeleton for the surface underlying the oven image into 3-space, by specifying the coordinates of each vertex, and representing the edges as straight-line segments; 2. to specify the tisting on every edge; 3. and to specify the colors of the threads for the edges. ithout needing to understand the underlying mathematics that e present here. Section 5 gives a concise definition of the EGRS model. Section 6 describes the mathematics that provides a formal description ho the graph rotation system model for specifying graph imbeddings, is extended to and re-interpreted as a robust model for specifying link structures immersed on orientable surfaces, hich thus induce oven images in 3D-space. After a brief look at knot theory, e sho by Theorem 6.3, our main theorem ho the EGRS model represents and operates on a cyclic plain-eaving structure, for hich all of the previously presented theory, in particular that for the edge surgery operations, becomes a necessity.

3 Extended Graph Rotation Systems as a Model for Cyclic Weaving 3 For general background on the mathematics of eaving, e recommend the classic paper of Grunbaum and Shephard [GrSh80], hich confines its attention to eaving in the plane. 2 Preliminaries on Topological Graph Theory Rather than presenting any ne results in topological graph theory, this section confines itself to some necessary background. It is consistent ith the discussions of these issues to be found in [GrTu87, BWGT09]. Alternative perspectives are provided by [MoTh01, Wh01]. 2.1 Graphs and surfaces Our graphs are alays undirected. Multiple edges ith the same endpoints and selfloops are alloed. A self-loop has only one endpoint, yet it has to distinguishable edge-ends. The distinction beteen the to edge-ends of a self-loop is achieved topologically by regarding the interior of each edge as parametrized by the open unit interval (0, 1). The edge-ends are small neighborhoods of the limit points 0 and 1, respectively, and, thus, they are distinguishable, even for a self-loop. This permits us to differentiate, for instance, beteen the to possible directions in hich one can traverse a self-loop. Any edge can be specified by a pair [v, ] of vertices, almost as one might do hen the discussion is confined to simple graphs. To adapt this form of specification to a multi-edge of multiplicity m, e can use m different names for one of their common endpoints and another m different names for the other endpoint. This satisfies our need for distinct edges ithin a multi-edge to have distinguishable names. When an edge is a self-loop, to names for the same endpoint v and may be interpreted, hen context requires, as the to distinguishable edge-ends of the edge. Under these conventions, each edge e = [v, ] induces to oriented edges v, and, v, each running from one edge-end of edge e to the other edge-end of e. We may sometimes denote the to oriented edges arising from e by e + and e. 1 A surface is a compact boundaryless 2-dimensional manifold, hich can be either orientable or non-orientable [Ch90, ChGrRi94, GrTu87]. An imbedding of a graph G on a surface S is a homeomorphism of the graph onto a topological subspace of the surface. It is alays assumed to be cellular, hich means that every connected component of S G, i.e., the interior of each face in the imbedding is homeomorphic to an open disk. Therefore, the surface S can be reconstructed, uniquely, from a graph G imbedded on the surface if the boundaries of all faces in the imbedding are specified it is achieved simply by pasting the perimeter of a proper polygon along each face boundary in the imbedding. In fact, it has been a common practice both in topological graph theory [GrTu87], as e describe in 2.2 and in computer graphics [Ba72] to represent surfaces along ith graph imbeddings. 2 1 We remark that e have adopted the concept of oriented edges from the literature of standard topological graph theory. It should be noted that our graphs are undirected so edges in a graph have no directions. Hoever, hen the face boundaries of a graph imbedding on a surface is traversed, it is critical to specify the order of the to ends of an edge in hich the traversing proceeds, hich involves the concept of oriented edges. 2 In computer graphics, graph imbeddings on surfaces are called meshes, and surface modeling based on graph imbeddings is called mesh-modeling [AkChSr03, ACSMT08, Ma88].

4 Extended Graph Rotation Systems as a Model for Cyclic Weaving Specifying imbeddings by rotation systems Graph rotation systems are used to give a purely combinatorial specification of an imbedding of a graph in a surface. Definition Let G be a graph ith n vertices. A rotation at a vertex v of G is a cyclic ordering of the oriented edges originating at v. A (pure) rotation system of the graph G consists of a set of n rotations, one for each vertex of G. Remark The spatial imagery in conceptualizing a rotation at a vertex v is that for each edge-end at v, there is an oriented edge that begins at that edge-end. Thus, a self-loop is associated ith to oriented edges in the rotation at v, representing the to respective directions in hich the self-loop could be traversed. It is easy to see that an imbedding π(g) of a graph G on an oriented surface naturally induces a pure rotation system of the graph G. Also, note that the faceboundary alk (for short, fb-alk) of each face in π(g) is a closed alk, that is, a (cyclically ordered) sequence of oriented edges. Conversely, it has been ell-knon since [He1891] and [Ed60] (for simple graphs) and [GrAl74] (for general graphs) that a pure rotation system ρ(g) of a graph G uniquely determines an imbedding of G on an oriented surface S, and, thus, uniquely determines the surface S. To do this, e need a face-tracing algorithm (see Algorithm 1, page 6. Note that for the pure rotation system ρ(g), all edges have tist-type 0). The face-tracing algorithm ill uniquely construct all fb-alks in ρ(g), from hich e can match each fb-alk of length s ith the perimeter of an s-sided polygon. This process, as explained above, ill uniquely reconstruct the surface S. A full exposition of rotation systems and face-tracing, ith examples, is given in 3.2 of [GrTu87]. Definition Given a rotation system ρ(g) for a graph G, a face corner (sometimes just corner) is a triple (v, e, e ), comprising a vertex v and to oriented edges e = v, u and e = v, u, both oriented out of v, here the v-edge-end of e immediately follos the v-edge-end of e in the rotation at v, as illustrated in Figure 2. If neither e nor e is a self-loop, e say that e is the 0-next to e at v and that e is the 1-next to e at v. For a self-loop, e must say hich orientation is 0-next or 1-next. u' e' v e Figure 2: The corner (v, e, e ), here e = vu and e = vu. Remark We emphasize that in this convention for specifying a corner, both of the oriented edges point out from v. When e rite (v, e, e ), it means that e follos e in the cyclic order at vertex v. In the special case of a vertex v of degree 2, there are corners (v, e, e ) and (v, e, e). u 2.3 Rotation systems and surgery Surgery operations on pure graph rotation systems (thus, on graph imbeddings in orientable surfaces) have been extensively studied [Ch90, GrTu87]. For example, inserting an edge into a pure graph rotation system adheres to the folloing rules:

5 Extended Graph Rotation Systems as a Model for Cyclic Weaving 5 Edge-Insert-0 If the to ends of an edge are inserted into to corners of the same face, then the edge splits that face into to faces, and the to oriented edges corresponding to the ne edge belong to the fb-alks of those to different faces in the ne imbedding. Figure 3 illustrates the splitting of a face ith boundary alk abcef into to faces, ith respective boundary alks abcd and d + ef. a v b d f Figure 3: Inserting the edge d into the face ith boundary alk abcef yields the faces abcd and d + ef. If the to ends of an edge are inserted beteen to corners of to different faces, then the inserted edge merges those to faces into a single face, and the to oriented edges corresponding to the ne edge belong to the fb-alk of that single face in the ne imbedding. Figure 4 illustrates the merger of to faces ith respective boundary alks abc and efgh into a single face, ith boundary alk abcd + efghd. c e u f b c e g d a Figure 4: Inserting the respective ends of edge d into to different faces merges them into a single face. The operation of edge-deletion on a pure rotation system reverses edge insertion. Edge-Delete-0 If the to oriented edges corresponding to an edge e appear in the fb-alks of to different faces, then deleting the edge e merges the to faces into a single face. This is illustrated in Figure 5. a v b d c u v d f e u Figure 5: When edge d appears in to different fb-alks, its deletion merges the rest of the to fb-alks (i.e., excluding d) into the fb-alk of a single face. If the to oriented edges corresponding to an edge e belong to the fb-alk of a single face, then deleting the edge e splits that face into to faces. This is illustrated in Figure 6. h a v b f c e u

6 Extended Graph Rotation Systems as a Model for Cyclic Weaving 6 a b d c h d g e f Figure 6: When edge d appears tice in a single fb-alk, its deletion splits the rest of that fb-alk (i.e., excluding d) into the fb-alks of to distinct faces. Definition A graph rotation system can be augmented to represent an imbedding on a non-orientable surface. For this, e label each edge [u, v] in the graph either as type-0 (flat) or type-1 (tisted), hich e call the tist-type of [u, v] and denote by t-type([u, v]). Definition A general rotation system of a graph G = (V, E) consists of a pure rotation system of G plus a mapping t-type : E {0, 1} that assigns to each edge of G a tist-type. It is also knon that an imbedding of a graph G on any surface (orientable or nonorientable) induces a general rotation system for the graph G, and that each general rotation system for the graph G induces an imbedding of G on an (orientable or non-orientable) surface. Moreover, the face-tracing algorithm can be generalized [GrTu87] to apply to general graph rotation systems. For convenience in our later discussion, e no present the General Face-Tracing Algorithm, hich is a modification of the algorithm described in [GrTu87], as Algorithm 1. A version of Algorithm 1 appears in [AkChGr09]. Algorithm 1 General Face-Tracing Algorithm. Subroutine FaceTrace(t 0, u 0, 0 ) Input: an oriented edge u 0, 0 ; and a number t 0 {0, 1}, called its trace-type. Remark: the trace-type corresponds to direction of traversal of the edge. Output: a sequence of oriented edges that forms an fb-alk containing u 0, trace and print u 0, 0 ; 2. t = t 0 + t-type([u 0, 0 ]) (mod 2); 3. u, = the t-next to 0, u 0 at 0 ; \\u = 0 4. hile ( u, u 0, 0 ) or (t t 0 ) do trace and print u, ; t = t + t-type([u, ]) (mod 2);, u =, u ; u, = the t-next to, u at. \\u = Algorithm FbWalks(ρ(G)) Input: ρ(g) is a general graph rotation system. Output: the collection of all fb-alks in ρ(g). hile there is an untraced face corner (u, e, e ) in ρ(g) do suppose that e = u, ; call FaceTrace(0, u, ). b a c e f h g Remark We observe that hereas face-tracing on pure rotation systems can stop hen all the oriented edges have been traced, face-tracing stops on general rotation systems hen all the face corners have been traced.

7 Extended Graph Rotation Systems as a Model for Cyclic Weaving 7 3 On the Edge-Tisting Operation Surgical edge operations have been poerful tools in the study of graph imbeddings in topological graph theory [GrTu87, MoTh01, Wh01] and in the research in computer graphics and the development of graphics softares [GuSt85, Ma88]. In particular, these operations offer the foundation for the development of a graphics mesh modeling system ith a highly interactive user-interface [AkChSr00, AkChSr03, ACSMT08]. Surgical edge operations on pure graph rotation systems have been extensively studied and ell understood. Hoever, hen edge-tists are introduced, certain fairly subtle phenomena occur. In fact, some of these intricacies seem to have not been addressed carefully in the literature of either topological graph theory or computer graphics. The underlying theory for our EGRS model is heavily based on edge-tists, and our graphics modeling system includes a highly interactive user-interface based on EGRS, hose development and analysis require a deep and thorough understanding of the objects and the surgical edge operations included ithin the EGRS model. In particular, the EGRS model for eaving regards an edge in the model as a thin strip, in hich the sides of the strip are regarded as to threads in the eave. Tisting an edge in the EGRS model specifies that these to threads ind around each other (see Section 6 for more details). This is quite different from the meaning of tisting an edge of a general rotation system in pure topological graph theory. We proceed ith a detailed, formal study of surgical edge operations on general graph rotation systems, from hich e can easily extend to the EGRS model. This section is focused on the edge-tisting operation, and Section 4 is devoted to the edge-insertion and edge-deletion operations. In particular, Theorem 3.4, Theorem 4.1, and Theorem 4.2 are ne theoretical results that repair imprecisions in hat has been described elsehere. Our terminology and notations here are a bit different from hat may be found elsehere. Throughout the discussion of Sections 3-4, e assume that ρ(g) is a general graph rotation system, and e consider the effect of applying an edgeoperation to an edge e in the graph G. Definition The operation of tisting an edge e in a general rotation system ρ(g) means changing its type, either from 0 to 1, or from 1 to 0. (When traversing a tisted edge during face-tracing, the cyclic direction at the terminating vertex at hich one selects the next oriented edge is taken to be opposite from the direction at the originating vertex.) 3.1 Calculating an fb-alk We closely examine the subroutine FaceTrace, as given ithin Algorithm 1. Definition Let t be the current trace-type and u, the next oriented edge to be traced, at some moment during the face-tracing process. The pair (t, u, ) is called a trace-pair. Note that the current trace-pair (t, u, ) uniquely determines the next tracetype t, by the rule t = t+type([u, ]) (mod 2); and then the trace-type t combined ith the oriented edge u, uniquely determines the next oriented edge, v to be traced: that is,, v is t -next to, u at. Therefore, any trace-pair (t, u, ) that occurs during the face-tracing process completely determines the fb-alk that is constructed by calling FaceTrace(t, u, ).

8 Extended Graph Rotation Systems as a Model for Cyclic Weaving 8 Lemma 3.1 Let h 1. Suppose that in a given general rotation system ρ(g) the first h trace-pairs traced by calling FaceTrace(t 0, u 0, u 1 ) are (t 0, u 0, u 1 ), (t 1, u 1, u 2 ),..., (t h 1, u h 1, u h ), and suppose that after tracing the h th oriented edge u h 1, u h, the process has trace-type t h. Then the first h trace-pairs traced by FaceTrace(t h, u h, u h 1 ), here t h = t h + 1 (mod 2), are (t h, u h, u h 1 ),..., (t 2, u 2, u 1 ), (t 1, u 1, u 0 ), here t i = t i + 1 (mod 2) for all i, h i 1, and after tracing the h th oriented edge u 1, u 0, the process has current trace-type t 0 = t (mod 2). Proof. The lemma can be easily verified for the case h = 1. We proceed by induction on h. Suppose that h > 1. For each i such that 0 i h, let τ i = t-type([u 0, u 1 ]) + + t-type([u i 1, u i ]). After tracing the i th oriented edge u i 1, u i, the process FaceTrace(t 0, u 0, u 1 ) has trace-type t i = t 0 + τ i (mod 2). Thus, t h = t 0 + τ h (mod 2), hich gives t h = t τ h (mod 2). No consider the process FaceTrace(t h, u h, u h 1 ). After tracing the first oriented edge u h, u h 1, the trace-type t n 1 of the process becomes (as e note that τ h + t-type([u h 1, u h ]) = τ h 1 (mod 2)) t h + t-type([u h 1, u h ]) = t τ h 1 = t h (mod 2). This proves that t h 1 = t h (mod 2). By the premises of this lemma, the tracing process FaceTrace(t 0, u 0, u 1 ) has tracetype t h 1 after tracing u h 2, u h 1, and the oriented edge u h 1, u h follos the oriented edge u h 2, u h 1 in that process. Thus, the oriented edge u h 1, u h must be t h 1 -next to the oriented edge u h 1, u h 2 at u h 1. In consequence, the edge u h 1, u h 2 must be t h 1 -next to u h 1, u h at u h 1. No since the process FaceTrace(t h, u h, u h 1 ) has a trace-type t h 1 after tracing the first oriented edge u h, u h 1, the next oriented edge to be traced by the process must be u h 1, u h 2. Therefore, after tracing the first oriented edge u h, u h 1, the process FaceTrace(t h, u h, u h 1 ) has a trace-pair (t h 1, u h 1, u h 2 ), and accordingly, it ill copy the process FaceTrace(t h 1, u h 1, u h 2 ) hen tracing the next h 1 trace-pairs. By the induction hypothesis, the first h 1 trace-pairs by the subroutine FaceTrace(t h 1, u h 1, u h 2 ) are ( t h 1, u h 1, u h 2 ),..., (t 2, u 2, u 1 ), (t 1, u 1, u 0 ), here t i = t i + 1 (mod 2) for all i such that h 1 i 1, and after tracing the (h 1) st oriented edge u 1, u 0, the process has trace-type t 0 = t (mod 2). Combining this ith the first traced oriented edge u h, u h 1, e conclude that the first h trace-pairs traced by the subroutine FaceTrace(t h, u h, u h 1 ) are (t h, u h, u h 1 ),..., (t 2, u 2, u 1 ), (t 1, u 1, u 0 ), here t i = t i + 1 (mod 2) for all i ith h i 1, and that after tracing the h th oriented edge u 1, u 0, the process has trace-type t (mod 2).

9 Extended Graph Rotation Systems as a Model for Cyclic Weaving Reversing an fb-alk Definition Suppose that the fb-alk B of a face F is the cyclically ordered sequence of trace-pairs (t 0, u 0, u 1 ), (t 1, u 1, u 2 ),..., (t h 2, u h 2, u h 1 ), (t h 1, u h 1, u 0 ). Then the reverse of the fb-alk B is defined to be the cyclically ordered sequence of trace-pairs B r = (t 0, u 0, u h 1 ), ( t h 1, u h 1, u h 2 ),..., (t 2, u 2, u 1 ), (t 1, u 1, u 0 ), here t i = t i + 1 (mod 2) for all i such that h 1 i 0. Corollary 3.2 Let (u 0, u 0, u h 1, u 0, u 1 ) be a face corner induced by a fixed general rotation system ρ(g) for a graph G. If the subroutine call FaceTrace(0, u 0, u 1 ) traces an fb-alk B, then the fb-alk traversed by the call FaceTrace(1, u 0, u h 1 ) is the reverse of B. Proof. First note that the last oriented edge on the fb-alk B traced by the subroutine call FaceTrace(0, u 0, u 1 ) must be u h 1, u 0, and that after tracing u h 1, u 0 the process must have trace-type 0 otherise, the process ould not have stopped. Therefore, e can assume that the fb-alk B is composed of the cyclically ordered trace-pairs (0, u 0, u 1 ), (t 1, u 1, u 2 ),..., (t h 2, u h 2, u h 1 ), (t h 1, u h 1, u 0 ). By Lemma 3.1, the first h trace-pairs by the subroutine FaceTrace(1, u 0, u h 1 ) are (1, u 0, u h 1 ), (t h 1, u h 1, u h 2 ),..., (t 2, u 2, u 1 ), (t 1, u 1, u 0 ), (3.1) here t i = t i +1 (mod 2), for all i such that h 1 i 1, and after tracing u 1, u 0, the trace-type is 1. Since u 0, u h 1 is 1-next to u 0, u 1 ) at u 0, the next edge to be traced by FaceTrace(1, u 0, u h 1 ) is u 0, u h 1, so the face-tracing process terminates. Therefore, the sequence in Eq (3.1) gives the entire fb-alk traced by the process FaceTrace(1, u 0, u h 1 ), hich is the reverse of B. Since the fb-alk B and its reverse bound the same topological region in the imbedding induced by the rotation system ρ(g), only one of them is needed. Therefore, the face corner (u 0, u 0, u h 1, u 0, u 1 ) is regarded as traced if the corner is traced either in the order u h 1, u 0, u 0, u 0, u 1, or in the order u 1, u 0, u 0, u 0, u h 1. Corollary 3.3 Fix a general rotation system ρ(g) and let [u 0, u 1 ] be an edge of G. Suppose that FaceTrace(t, u 0, u 1 ) traces an fb-alk B. Then the fb-alk traced by calling FaceTrace(t, u 1, u 0 ), here t = t type([u 0, u 1 ]) (mod 2), is the reverse alk B r. Proof. First suppose that t = 0. Consider the face corner (u 0, u 0, u h 1, u 0, u 1 ) in ρ(g). As discussed in the proof of Corollary 3.2, the fb-alk B traced by FaceTrace(0, u 0, u 1 ) can be assumed to be (0, u 0, u 1 ), (t 1, u 1, u 2 ),..., (t h 2, u h 2, u h 1 ), (t h 1, u h 1, u 0 ). Using Corollary 3.2, e infer that the fb-alk traced by FaceTrace(1, u 0, u h 1 ) is (1, u 0, u h 1 ), ( t h 1, u h 1, u h 2 ),..., (t 2, u 2, u 1 ), (t 1, u 1, u 0 ), hich e observe is B r, the reverse of B. Immediately before tracing u 1, u 0, the process FaceTrace(1, u 0, u h 1 ) has trace-type t 1 = 1 + type([u 0, u h 1 ]) + + type([u 2, u 1 ]) = 1 + type([u 1, u 0 ]) (mod 2).

10 Extended Graph Rotation Systems as a Model for Cyclic Weaving 10 Therefore, the fb-alk B r is also produced by the process FaceTrace(t 1, u 1, u 0 ), here t 1 = 1 + type([u 0, u 1 ]) = t type([u 0, u 1 ]) = t (mod 2) This proves that FaceTrace(t, u 1, u 0 ) also traces the reverse B r of the fb-alk B. Alternatively, suppose that t = 1. In this case the face corner to be considered is (u 0, u 0, u 1, u 0, u h 1 ). Here, too, it can be easily seen that the fb-alk B traced by FaceTrace(1, u 0, u 1 ) can be assumed to be of the form (1, u 0, u 1 ), (t 1, u 1, u 2 ),..., (t h 2, u h 2, u h 1 ), (t h 1, u h 1, u 0 ). Corollary 3.2 implies that the fb-alk traced by FaceTrace(0, u 0, u h 1 ) is (0, u 0, u h 1 ), (t h 1, u h 1, u h 2 ),..., (t 2, u 2, u 1 ), (t 1, u 1, u 0 ). (3.2) and that FaceTrace(0, u 0, u h 1 ) traces the reverse B r of the fb-alk B. (Note that the reverse of the reverse of a alk B is the alk B itself). Therefore, immediately before the process FaceTrace(0, u 0, u h 1 ) traces the last oriented edge u 1, u 0 in the trace-pair sequence (3.2), the trace-type is t 1 = type([u 0, u h 1 ]) + + type([u 2, u 1 ]) (mod 2) = type([u 0, u 1 ]) = t type([u 0, u 1 ]) (mod 2) = t. Therefore, the process FaceTrace(t, u 1, u 0 ) and the process FaceTrace(0, u 0, u h 1 ) trace the same fb-alk B r, hich is the reverse of B. Accordingly, for each oriented edge u,, there are only to essentially different trace-pairs. By Corollary 3.3, the four trace-pairs (0, u, ), (1, u, ), (0,, u ) and (1,, u ) of an edge [u, ] correspond to only to different trace-pairs used in the face tracing process Facing(ρ(G)). These trace-pairs ill be called the tracepairs induced on the edge [u, ]. The trace-pairs (0, u, ), (1, u, ) are said to use the oriented edge u,, and the trace-pairs (0,, u ) and (1,, u ) are said to use the oriented edge, u. 3.3 Effects of edge-tisting surgery Theorem 3.4 Tisting an edge e in a general rotation system ρ(g) satisfies the folloing rules: (A) Suppose that the to trace-pairs induced by edge e belong to the fb-alks of to different faces in the imbedding. Then tisting edge e merges the to faces into a single face; (B) Suppose that the to trace-pairs induced by edge e belong to the fb-alk of the same face F in the imbedding. (B1) If the to trace-pairs induced by e use the same oriented edge, then tisting edge e splits the face F into to faces; (B2) If the to trace-pairs induced by edge e use different oriented edges, then tisting edge e converts the face F into a ne single face.

11 Extended Graph Rotation Systems as a Model for Cyclic Weaving 11 Proof. Let e = [u, ]. We first consider case (A). By Corollary 3.3, e can assume that the to faces hose fb-alks contain the trace-pairs induced by e are F 0 = FaceTrace(0, u, ) : F 1 = FaceTrace(1, u, ) : (0, u, ), (t 0,, 0 ),..., (t 0, x 0, u ), and (1, u, ), (t 1,, 1 ),..., (t 1, x 1, u ), here e have t 0 + t 1 = 1. Note that no trace-pairs induced by the edge [u, ] can appear anyhere else except the to places given in the above sequences. See the top part of Figure 7(1) for an illustration x 2 1 x 1 F 1 F 0 1 u x u 0 2 x 2 u x 1 1 x 1 u x 0 u 2 u x 1 (1) Case (A) (2) Case (B1) (3) Case (B2) Figure 7: Tisting an edge in a general rotation system No e tist the edge e (i.e., changing the tist-type, as shon at the bottom of Figure 7(1)), and e thereby obtain a ne rotation system ρ (G) of the graph G. We apply FaceTrace(t 0,, 0 ) to ρ (G). Since no vertex rotation is changed and no other edge changes the tist-type, FaceTrace(t 0,, 0 ) on ρ (G) follos the same sequence of fb-alk F 0 (starting from (t 0,, 0 )), until it traces the tracepair (0, u, ). No since the edge e = [u, ] changes tist-type, after tracing (0, u, ), the trace-type becomes t = t 1 (mod 2). From the sequence for F 1, e kno that, 1 is the t 1 -next to, u at. Therefore, the next oriented edge to be traced by FaceTrace(t 0,, 0 ) on ρ (G) should be, 1 (see the bottom of Figure 7(1)). Thus, after tracing the trace-pair (0, u, ), the process FaceTrace(t 0,, 0 ) on ρ (G) has a trace-pair (t 1,, 1 ). Again since no vertex rotation is changed and no other edge changes the tist-type, FaceTrace(t 0,, 0 ) no ill follo the sequence F 1 (starting from (t 1,, 1 )) until it traces the tracepair (1, u, ). By an analysis similar to that above, after tracing (1, u, ), e see that the process FaceTrace(t 0,, 0 ) has a trace-type t 0 and that the next edge to be traced is, 0 (see the bottom of Figure 7(1)). Therefore, the process closes its tracing of the fb-alk. Since the process FaceTrace(t 0,, 0 ) on ρ (G) uses exactly the same trace-pairs used in fb-alks F 0 and F 1, and since no vertex rotation is changed and no other edge changes tist-type, e conclude that the set of faces for the rotation system ρ (G) can be obtained from that for ρ(g) by replacing the fb-alks F 0 and F 1 in ρ(g) by a single fb-alk produced by FaceTrace(t 0,, 0 ) on ρ (G). This completes the proof for case (A).

12 Extended Graph Rotation Systems as a Model for Cyclic Weaving 12 We next consider case (B1). By the premise, e may assume that the fb-alk F is (see the top of Figure 7(2)) F : (t 0,, 0 ),..., (0, u, ), (t 1,, 1 ),..., (1, u, ). Note that in the above sequence, e have t 0 + t 1 = 1. Tisting the edge e = [u, ] gives a ne rotation system ρ (G) of the graph G. We apply FaceTrace(t 0,, 0 ) to ρ (G). By the same reasoning as that in (A), the process follos the sequence of F (starting from (t 0,, 0 )) until it traces the middle trace-pair (0, u, ). Since the tist-type of edge [u, ] has changed, the trace-type after tracing the edge u, is t = t 0 (mod 2). In the sequence of fb-alk F, edge, 0 is t 0 -next to edge, u at. Therefore, the process FaceTrace(t 0,, 0 ) on ρ (G) closes its tracing ith the fb-alk (see the bottom of Figure 7(2)) F 0 : (t 0,, 0 ),..., (0, u, ). Completely similarly, e observe that applying FaceTrace(t 1,, 1 ) on ρ (G) ill result in the fb-alk F 1 : (t 1,, 1 ),..., (1, u, ). Since the trace-pairs used by F 0 and F 1 are exactly those used by F, e conclude changing the tist-type of edge e splits the face F into to faces F 0 and F 1. Finally e discuss case (B2), hich is not mentioned in [GrTu87] or elsehere. By the premise, e may assume that the fb-alk of F is (see the top of Figure 7(3)) F : (0, u, ), (t 1,, 1 ),..., (t, x 2, ), (t 1,, u ), (0, u, 2 ),..., (t, x 1, u ), here the trace-types t and t are irrelevant, and e have let t 1 = type([u, ]). We define t 0 = t (mod 2). Note that the trace-type before the oriented edge, u in the sequence is also t 1 : by Corollary 3.3, the trace-pairs (0, u, ) and (t 0,, u ) are equivalent. Within the sequence F, e observe the folloing: (b1). u, is 0-next to u, x 1 at u; (b2). u, 2 is 0-next to u, at u; (b3)., x 2 is t 0 -next to, u at ; (b4)., u is t 0 -next to, 1 at ; here (b3) follos because, u is t 1 -next to, x 2 at, and (b4) because, 1 is t 1 -next to, u at. From the subsequence F : (t 1,, 1 ),..., (t, x 2, ), t 1 in F (here the last t 1 is the trace-type after tracing F ), by Lemma 3.1, FaceTrace(t 0,, x 2 ) ill produce the reverse of F as F r : (t 0,, x 2 ),..., (t, 1, ), t 0 (here the trace-type t is irrelevant). We tist the edge [u, ] (i.e., changing the tist-type of [u, ] to t 0 ), and e apply FaceTrace(0, u, ) to the ne rotation system ρ (G). After tracing (0, u, ), the trace-type becomes t 0. By (b3), the next oriented edge to be traced by the process is, x 2. Therefore, the modified process ill mimic the process FaceTrace(t 0,, x 2 ) on the original rotation system ρ(g) and produce the subsequence F r : (t 0,, x 2 ),..., (t, 1, ), t 0, until it returns to the vertex from 1,. Note that after tracing 1,, the process has a trace-type t 0. No by (b4), the process continues tracing the tracepair (t 0,, u ) and changes its trace-type to t 0 + t 0 = 0 (mod 2). By (b2), the next

13 Extended Graph Rotation Systems as a Model for Cyclic Weaving 13 oriented edge to be traced thereafter is u, 2. Hence, the process then follos the sequence of fb-alk F (starting from (0, u, 2 )) until it traces the trace-pair (t, x 1, u ) after tracing x 1, u, the process has a trace-type 0. By (b1), the next edge to be traced is u,. Therefore, the process FaceTrace(0, u, ) on ρ (G) produces the fb-alk F, given by (0, u, ), (t 0,, x 2 ),..., (t, 1, ), (t 0,, u ), (0, u, 2 ),..., (t, x 1, u ). hich uses exactly the same oriented edges as fb-alk F. In conclusion, tisting the edge [u, ] in this case replaces the face F by a single ne face F. Theorem 3.4 is consistent ith hat is described by [St78] and [Zh96]. Results on the set of fb-alks of tisting an edge hose to induced trace-pairs both belong to the same fb-alk, that is, case (B2) of Theorem 3.4, seem less than explicit in the existing literature on topological graph theory. 4 On Edge-Insertion and Edge-Deletion Edge insertion/deletion operations have been fundamental in the studies of pure graph rotation systems on orientable imbeddings [Ch90, GrTu87] and in their applications to computer graphics [AkCh99]. One might expect that most corresponding results involving insertion and deletion operations on pure graph rotation systems ould extend naturally to general graph rotation systems. Hoever, there seem to be certain subtle issues that are quite different, and hich, to our knoledge, have not been thoroughly studied in the literature. In particular, the parts of Theorems 4.1 and 4.2 concerned ith Case (B2) are ne. Accordingly, e present a revised study of those operations on general graph rotation systems in this section. To observe ho the surgery operations of edge-insertion and edge-deletion are changed, e consider the general graph rotation systems hose rotation projections [GrTu87] are given in Figure 8 (here the oriented edges are assigned counterclockise order, and the edges of type-1 are marked by a cross ). Figure 8(1) corresponds to a one-face imbedding of the bouquet B 1 (one vertex ith one selfloop) on the projective plane. In particular, the face corners c 1 and c 2 in Figure 8(1) belong to the same face. No suppose that e insert a ne type-0 edge e 2 beteen these to face corners, as depicted in Figure 8(2). Our rules in 2 for pure graph rotation systems say that an edge insertion (necessarily type-0 for pure rotation systems) beteen to corners of the same face ould split that face into to faces. Hoever, hen applying the face tracing procedure in Algorithm 1 to the graph of Figure 8(2), e find out that the resulting rotation system corresponds to a one-face imbedding of the bouquet B 2 (on the Klein bottle)! e 1 c 2 c1 e 2 e 1 (1) Figure 8: Inserting an edge into a general rotation system This phenomenon seems never explicitly described in the literature. A careful examination indicates that hen tracing the face in Figure 8(1), the to face corners (2)

14 Extended Graph Rotation Systems as a Model for Cyclic Weaving 14 c 1 and c 2 are traversed in opposite directions (if corner c 1 is traversed in counterclockise order then corner c 2 ould be traversed in clockise order). Motivated by our observation of this phenomenon, e introduce the folloing ne concept. Definition Let ρ(g) be a general rotation system for a graph G. Each face corner (v, e, e ) in ρ(g) is assigned a corner-type by the algorithm FbWalks(ρ(G)), as follos: (1). if FbWalks(ρ(G)) passes through vertex v by entering along edge e and leaving along edge e, then the corner (v, e, e ) has corner-type 0; (2). if FbWalks(ρ(G)) passes through vertex v by entering along edge e and leaving along edge e, then the corner (v, e, e ) has corner-type 1; We observe that if ρ(g) is a pure graph rotation system, then e can make all face corners type-0. Remark The corner-types depend on ho the algorithm call FbWalks(ρ(G)) picks an untraced corner (u, e, e ) to start tracing the fb-alk for each ne face F. There are only to ays to assign a first corner-type for a ne face F : one is by FaceTrace(0, e ) and the other by FaceTrace(1, e). By Corollary 3.2, the to ays to assign the first corner-type lead to exactly opposite corner types at every corner. The point is, for any to corners of F, either both ays assign the to corners the same type, or both ays assign the to corners different types. Therefore, since our concern is only hether they have the same or diferent types, it does not matter hich ay is used to assign the first corner-type for a face. Our chief concern here is ho surgery operations on a general graph rotation system impact the corresponding imbedding on a surface. At the outset of this section, e observed that this correspondence is non-obvious. In fact, for edge-insertions into general graph rotation systems, the rules given by the folloing theorem differ significantly from those for pure graph rotation systems. Since inserting a type-1 (i.e., tisted) edge can be implemented by inserting a type-0 (i.e., flat) edge folloed by tisting the edge, e ill consider here only the case of inserting type-0 edges. 4.1 Effects of edge-insertion surgery Theorem 4.1 Suppose that e insert the ends of a type-0 edge e into to face corners c 1 and c 2 in a general rotation system ρ(g). Then the folloing rules hold: (A) If the corners c 1 and c 2 belong to to different faces, then inserting edge e beteen c 1 and c 2 merges the to faces into a single face. (B) If the corners c 1 and c 2 belong to the same face, then (B1) hen c 1 and c 2 have the same corner-type, then inserting edge e beteen c 1 and c 2 splits the face into to faces; (B2) hen c 1 and c 2 have different corner-types, then inserting e beteen c 1 and c 2 results in a ne face. Proof. In case (A), since the corners c 1 and c 2 belong to different faces, the remark given just above this theorem allos us to assume that both corners have type-0. This reduces case (A) to inserting a type-0 edge beteen type-0 face corners of to different faces, hich e kno merges those to faces into a single face. For case (B1), since the corners c 1 and c 2 have the same type, the same remark permits us to assume again that both corners have type-0. This reduces case (B1)

15 Extended Graph Rotation Systems as a Model for Cyclic Weaving 15 to the case of inserting a type-0 edge beteen to type-0 corners of the same face, hich e kno splits that face into to faces. A proof of cases (A) and (B1) for pure rotation systems appears in [FuGrMc88] ( 2.4, page 526). For case (B2), e may assume that the corner c 1 = (v 1, v 1, 1, v 1, u 1 ) of face F is type-0, and that the corner c 2 = (v 2, v 2, u 2, v 2, 2 ) of face F is type-1, by that same remark. Suppose that the ne edge [v 1, v 2 ] is inserted beteen corners c 1 and c 2. Under these assumptions, the fb-alk of the face F of the original rotation system ρ(g) is of the form F : (, 1, v 1 ), (0, v 1, u 1 ), α, (, 2, v 2 ), (1, v 2, u 2 ), β, here represents trace-types that are irrelevant, and here α and β are subsequences in the fb-alk. Note that v 1, u 1 is 0-next to v 1, 1 at v 1 and that v 2, u 2 is 1-next to v 2, 2 at v 2. See Figure 9(1) for an illustration. After inserting the ne edge [v 1, v 2 ], the resulting rotation system ρ (G ) (here G is G plus the ne edge) has the folloing ne relations (see Figure 9(2)): (b1) v 1, v 2 is 0-next to v 1, 1 at v 1 ; (b2) v 1, u 1 is 0-next to v 1, v 2 at v 1 ; (b3) v 2, v 1 is 1-next to v 2, 2 at v 2 ; (b4) v 2, u 2 is 1-next to v 2, v 1 at v 2. u 1 u 2 v 1 v u 1 u 2 v 1 v β (1) α Figure 9: Inserting an edge: case (B2) β r (2) No consider the tracing process π = FaceTrace(0, v 1, u 1 ) in the ne rotation system ρ (G ). Since no other face corners are affected by the edge insertion, the process π follos the sequence of F (starting from (0, v 1, u 1 )) until it traces the oriented edge 2, v 2 : S 1 : (0, v 1, u 1 ), α, (, 2, v 2 ), 1, here the last 1 indicates the trace-type after tracing the oriented edge 2, v 2. Then because of (b3), the next oriented edge to be traced by π is v 2, v 1, and after tracing v 2, v 1, the trace-type is 1 (because the ne edge [v 1, v 2 ] has tist-type 0): S 2 : (1, v 2, v 1 ), 1. By relation (b1), the next oriented edge to be traced by π is v 1, 1. By Corollary 3.2, in the original rotation system ρ(g), the process FaceTrace(1, v 1, 1 ) ill trace the reverse of FaceTrace(0, v 1, u 1 ) (hich is F ). Therefore, the process π ill follo the reverse of F (starting from (1, v 1, 1 ) and changing all corner types) until it traces the oriented edge u 2, v 2 : S 3 : (1, v 1, 1 ), β r, (, u 2, v 2 ), 0, here β r is the reverse of β. Note also that by Lemma 3.1, after tracing u 2, v 2, the process π has a trace-type 0. α

16 Extended Graph Rotation Systems as a Model for Cyclic Weaving 16 By relation (b4), the next oriented edge to be traced by the process π is v 2, v 1, and after tracing v 2, v 1, the process π has a trace-type 0 (because the edge [v 1, v 2 ] has type-0): S 4 : (0, v 2, v 1 ), 0. No by relation (b2), the next edge must be v 1, u 1. It follos that the process π = FaceTrace(0, v 1, u 1 ) terminates and completes its tracing ith an fb-alk that is the concatenation of the sequences S 1, S 2, S 3, and S 4 above, i.e., e obtain the sequence F = (0, v 1, u 1 ), α, (, 2, v 2 ), (1, v 2, v 1 ), (1, v 1, 1 ), β r, (, u 2, v 2 ), (0, v 2, v 1 ). Note that the face F has used both trace-pairs for the ne edge [v 1, v 2 ] (i.e., (0, v 2, v 1 ) and (1, v 2, v 1 ) in the above sequence) plus the same trace-pairs used in F. (By Corollary 3.3, e may say that a trace-pair (t, v, ) is used if either (t, v, ) is used or (t,, v ) is used, here t = t + t-type([v, ]) + 1 (mod 2)). No other face corners are affected, and e conclude that the ne set of fb-alks in case (B2) can be obtained from those for the original rotation system ρ(g) by replacing face F by the ne face F. This completes the proof for case (B2). The results of edge-insertion beteen face corners of different types in a general rotation system, i.e., case (B2) in Theorem 4.1, seem not to have been explicated heretofore in the literature. 4.2 Effects of edge-deletion surgery No e turn to edge-deletion on general rotation systems. Since deleting an edge e of tist-type 1 can be implemented by first tisting that edge and then deleting the resulting edge e of tist-type 0, it is sufficient to focus on deleting a type-0 edge. Theorem 4.2 Deleting a type-0 edge e from a general graph rotation system ρ(g) satisfies the folloing rules: (A) Suppose that the to trace-pairs induced by edge e belong to the fb-alks of to different faces of the imbedding. Then deleting edge e merges the to faces into a single face. (B) Alternatively, suppose that the to trace-pairs induced by e both belong to the fb-alk of the same face F in the imbedding. (B1) If the to trace-pairs induced by edge e use different oriented edges, then deleting edge e splits the fb-alk of the face F into to closed alks, each of hich is the fb-alk of a ne face of the resulting imbedding. (B2) If the to trace-pairs induced by edge e = [u, ] use the same oriented edge, then deleting edge e changes the fb-alk of face F into the fb-alk of a single ne face. Proof. As in Theorem 4.1, proofs for cases (A) and (B1) can be derived by modifying the proofs of the corresponding theorem for edge-deletion from a pure rotation system. See the rule Edge-Delete-0 in 2. Thus, e need to prove the theorem only for case (B2). By the premises, e can assume that the fb-alk of face F is F : (0, u, ), (0,, 0 ), α, (, u 1, u ), (1, u, ), (1,, 1 ), β, (, u 0, u ) as shon in Figure 10(1) (note that [u, ] has tist-type 0), here α and β are subsequences in the fb-alk and stands for irrelevant trace-types.

17 Extended Graph Rotation Systems as a Model for Cyclic Weaving 17 β α u1 u u 0 β r α u1 u u 0 (1) (2) Figure 10: Deleting an edge: case (B2) In the above sequence, e have assumed the folloing: (b1) u, is 0-next to u, u 0 at u; (b2), 0 is 0-next to, u at ; (b3) u, is 1-next to u, u 1 at u; and (b4), 1 is 1-next to, u at. Therefore, after deleting the edge [u, ] (see Figure 10(2)), (b5), 0 becomes 0-next to, 1 at, by (b2) and (b4), and (b6) u, u 0 becomes 1-next to u, u 1 at u, by (b1) and (b3). Let ρ (G ) be the rotation system obtained by deleting edge [u, ] from ρ(g), here G is the graph G [u, ]. The tracing process π = FaceTrace(0,, 0 ) follos the fb-alk of face F (starting from (0,, 0 )) until it traces the trace-pair (, u 1, u ): S 1 : (0,, 0 ), α, (, u 1, u ), 1. The last 1 is the trace-type after tracing the subsequence S 1. By (b6), the oriented edge u, u 0 is 1-next to u, u 1 at u. Therefore, the next oriented edge to be traced is u, u 0, and the process π follos the sequence produced by FaceTrace(1, u, u 0 ). By Lemma 3.1, this process ill follo the reverse of F (starting from (1, u, u 0 )) until it traces the oriented edge 1, : S 2 : (1, u, u 0 ), β r, (, 1, ), 0, here β r is the reverse of the subsequence β. After tracing the subsequence S 2, the trace-type is 0. No by (b5), the next edge to be traced is, 0. Therefore, the process π =FaceTrace(0,, 0 ) terminates ith the fb-alk that is the concatenation of S 1 and S 2 : F : (0,, 0 ), α, (, u 1, u ), (1, u, u 0 ), β r, (, 1, ). Since the face F uses all trace-pairs of F except for the to induced by [u, ], and since [u, ] has been deleted from ρ(g), e conclude that the rotation system ρ (G ) can be obtained from the rotation system ρ(g) by replacing the face F by the face F. This completes the proof for case (B2). As ith Theorem 4.1, prior literature does not seem to include a proof of case (B2) in Theorem Extended Graph Rotation Systems The extensions to the edge-tisting operation that are no to be described are crucial for our approach to modeling cyclic eaving on orientable surfaces. They are for specifying eaves, and not for graph imbeddings on surfaces.

18 Extended Graph Rotation Systems as a Model for Cyclic Weaving 18 Whereas labeling each edge ith a number 0 or 1 is sufficient for specifying graph imbeddings, our edge-labels for modeling eaving are dran from the entire set of integers. Whereas in classical topological graph theory, an edge is either untisted or tisted, in the EGRS model, by ay of contrast, its states ith respect to tisting are in bijective correspondence ith the integers. The EGRS model as introduced in [AkChGr09]. What appears in this section is essentially a reprise. In a topological understanding of graph theory, tracing a tisted edge reverses the local orientation of the rotation system; accordingly, the result of double-tisting an edge is topologically equivalent to an untisted edge. By ay of contrast, in our model for cyclic eaving, the to trace-pairs induced by a type-0 edge are regarded as to parallel segments and tisting the edge is interpreted as crossing the to segments. We are also interested in knoing hich segment goes over and hich one under at the crossing point, and by ho many turns one segment is tisted around the other. In our model of cyclic eaving, double tisting an edge is not the same as leaving it untisted. Figure 11 gives some intuitive illustrations for edge-tisting in terms of the above interpretation. (1) (2) (3) (4) (5) k =0 k =1 k = 1 k =2 k = 2 Figure 11: (1) an untisted edge; (2) a clockise tisted edge; (3) a counterclockise tisted edge; (4) a double-clockise tisted edge; (5) a doublecounterclockise tisted edge. Comparing (2) ith (3) and (4) ith (5) in Figure 11 reveals that the direction in hich e tist the edge is clearly relevant to hich segment passes over the other. Motivated by this, e introduce the folloing definitions. Definition An edge is 1 + -tisted (resp. 1 -tisted) if it is obtained from a flat rectangular paper strip, in hich the to longer sides of the rectangle are interpreted as the to trace-pairs induced by the edge, by fixing one end of the strip and tisting the other end in clockise (resp. counterclockise) direction by 180. See Figure 11(2)-(3) for an illustration. Note that this definition is independent of hich end and hich side of the paper strip is fixed. More generally, e say that an edge is k + -tisted (resp. k -tisted) for an integer k 0 if the edge can be obtained from an untisted edge by k consecutive 1 + -tists (resp. 1 -tists). Definition An extended rotation system for a graph G is obtained from a pure rotation system by assigning a number k of tists, ith k Z, to every edge of G. We point out that in terms of graph imbeddings on surfaces, as studied in [GrTu87, GuSt85], a k + -tisted or k -tisted edge is equivalent to an untisted edge if k is even, and equivalent to a normally tisted edge if k is odd.

19 Extended Graph Rotation Systems as a Model for Cyclic Weaving 19 6 Cyclic Plain-Weaving on Surfaces We no examine ho graph rotation systems can be used to create oven images on orientable surfaces. In the rest of this paper, surfaces ill alays refer to orientable surfaces, unless explicitly specified otherise. We begin ith some precise definitions. Theorem 6.3, hich provides the theoretical basis for cyclic plain-eaving, is the main ne result of this paper. Definition A cyclic eaving on a surface S is an immersion σ : C S of the union of a finite set C = {c 1,..., c k } of circles. We require that every point in the disjoint union has a neighborhood that is mapped homeomorphically. Hoever, the images of to circles in C may intersect on the surface S, and the image of a single circle may self-intersect. The number of intersection points of the circles on the surface is finite, and the number of pre-images of any point on S under σ is finite. Moreover, every intersection is a true intersection, rather than a tangency. Definition The thickness of a cyclic eaving σ is the maximum number of preimages of a point p in the surface S under the mapping σ, taken over all p σ(c). Moreover, e assume that for each point p in the image σ(c), the mapping σ also specifies an ordering of the pre-images of p, so that for to circles ho images intersect at p e kno hich is over the other at the point p. We ill concentrate on a most common cyclic eaving structure, defined as follos: Definition A cyclic eaving σ : C S is a cyclic plain-eaving if it satisfies the folloing conditions: Its thickness is at most 2; For each circle c C, suppose that p 1, p 2,..., p m is the ordered sequence of crossing points encountered hen traversing σ(c) on S. Then σ(c) assigns these points alternatingly as over and under. That is, the closed curve σ(c) eaves alternatingly over and under as it crosses other circles or itself. We point out that there are also other popular eaving patterns, such as tilleaving and satin-eaving [GrSh80]. In fact, our recent research (e.g., [ACCXG11]) shos that our EGRS model can also be used for these eaving structures. 6.1 The eaving genus of a link A knot is defined as an imbedding of the circle S 1 in R 3. A link is an imbedding of a disjoint union of one or more circles in 3-space R 3, although usually the ord knot is used hen there is only one circle. In classical knot theory (e.g., see [Ad04, CrFo77, Mu08]), a knot or link is commonly represented by a projection onto the plane R 2, as illustrated in Figure 12. The central concern of knot theory is deriving ays to decide from their representations hether to knots (or to links) are equivalent. That is, could one of them be deformed into the other? (a) (b) Figure 12: (a) A trefoil knot. (b) The Hopf link.

20 Extended Graph Rotation Systems as a Model for Cyclic Weaving 20 For the case in hich the surface S is a sphere, a cyclic plain-eaving on S is equivalent to hat topologists call an alternating projection of a link. (See [Ad04].) To a topologist, the link itself is an imbedding of the set C of circles in 3-space, or more formally, an equivalence class of imbeddings that are ambient isotopic to each other. As far as e kno, the notion of alternating projections onto surfaces other than the sphere has not been ell-developed by topologists. We briefly digress from eaving into some link theory. Proposition 6.1 For any link L in 3-space R 3 ith n components c 1,..., c n, there is a closed orientable surface of genus n in R 3 on hich L is imbedded. Proof. Thicken each component c j into a solid torus, so that c j lies on the surface of that solid torus, and so that the solid tori are mutually disjoint. Next discard the interiors of the solid tori, so that each component of the link lies on a torus. Then connect the n tori ith n 1 tubes, to obtain a copy S of the surface S n of genus n. Remark A closed surface in R 3 separates R 3 into to parts, by a 3-dimensional analogue of the Jordan curve theorem. The part that extends to infinity is called the outside, and the other part is called the inside. Definition Restoration of a link L from a projection onto a surface is the result of pulling each crossing apart: a small over-crossing segment is pulled outside the surface and a small undercrossing segment is pushed inside the surface. Definition An alternating projection of a link L in R 3 onto a surface S is a continuous function R 3 S hose restriction to L is a cyclic plain-eaving, hose restoration is equivalent to L in R 3. Corollary 6.2 Every link L in 3-space has an alternating projection. Proof. By Proposition 6.1, there is a closed orientable surface S in R 3 such that link L is imbedded on S. An imbedding is an alternating projection ith zero crossings. Definition The eaving genus of a link L in R 3 is the minimum genus taken over all closed surfaces in R 3 onto hich the link L has a cyclic plain-eaving. Remark For any integers n and g such that n 1 and 0 g n, it is possible to construct an n-component link ith eaving genus g. Use g copies of a non-trivial knot K and n g copies of the trivial knot, such that each component is splittable from the others. 6.2 Constructing plain-eavings Let π 0 : G S be an imbedding of a graph on an oriented surface, and let ρ 0 (G) be the corresponding pure rotation system, hose rotations are taken to be counterclockise around the vertices. We no present a method to construct a cyclic plain-eaving on S. From a topological perspective, this method could be described in terms of surgery on a regular neighborhood of the image of the graph G in the surface S, in the form of a band-decomposition [GrTu87] of S. A combinatorial specification is essential for a computer-graphics implemenation. Each edge [u, ] of G corresponds to to trace-pairs (0, u, ) and (1, u, ) in ρ 0 (G) (as ell as in π 0 (G)). Note that by Corollary 3.3, these to trace-pairs

21 Extended Graph Rotation Systems as a Model for Cyclic Weaving 21 have to equivalent trace-pairs in terms of the oriented edge, u. In general, for each trace-pair (t 1, u, ), e create a corresponding segment t1 u, t2, here t 2 = t 1 + t-type([u, ]) (mod 2) is the trace-type e obtain if e start ith the trace-type t 1 and trace the oriented edge u,. The endpoints u and are called the head and the tail of the segment, respectively. To obtain a cyclic eaving σ 0 on the surface S, e proceed as follos. For each edge [u, ] of ρ 0 (G), e create the to segments 0 u, 0 and 1 u, 1 on S (note that ρ 0 (G) is a pure rotation system). For the fb-alk of a face F : (0, u 1, u 2 ),..., (0, u h 1, u h ), (0, u h, u 1 ), (6.1) e let the tail of the segment 0 u i 1, u i 0 be connected to the head of the segment 0 u i, u i+1 0 for all i = 2,..., h + 1 (here e let h + 1 = 1 and h + 2 = 2). Remark We visualize the segment 0 u, 0 as lying slightly to one side of the edge [u, ] and the segment 1 u, 1 as lying slightly to the other side, as illustrated in Figure 13. 1, y 1 u, 1 y u x 0 u, 0 (A1) 0, x 1, y 1 u, 1 y u x 0 u, 0 (A2) 0, x 1 u, 1 u 0 u, 0 (A3) 1, y y x 0, x 1 u, 0 u 0 u, 1 (A4) 1, y y x 0, x 1 u, 0 u 0 u, 1 (A5) 1, y y x 0, x u 1 u, 1 0 u, 0 1 u, 1 u 0 u, 0 (B1) (B3) 1, y y x 0, x 1, y y x 0, x u 1 u, 1 0 u, 0 1 u, 1 u 0 u, 0 (B2) (B4) 1, y y x 0, x 1, y y x 0, x Figure 13: Weaving structure changes corresponding to 1 + -tist and 2 + -tist This is consistent ith hat e ould obtain if e used the reverse of the fb-alk of F and the segments of the form 1 u i, u i 1 1. Thus, the face F contains a circle c F composed of the segments: c F : 0 u 1, u 2 0,,..., 0 u h 1, u h 0, 0 u h, u 1 0. (6.2)

22 Extended Graph Rotation Systems as a Model for Cyclic Weaving 22 We visualize the circle c F as lying near the boundary of the face F, just slightly into the interior of F. We construct such segments and circles in the surface S for every fb-alk in ρ 0 (G), so that hen e traverse an edge [u, ] from u to on the surface S, the segment 0 u, 0 is to our right side and the segment 1 u, 1 to our left side. This is consistent ith having the rotations in counterclockise order. This yields an initial eaving σ 0 from the circle collection to the surface S: C 0 = {c F F is a face in π 0 (G)} Notation We use the symbol to designate trace-types hose values make no difference to the discussion. We no describe ho to change the cyclic eaving structure σ 0 corresponding to k + - tisting an edge [u, ] in the rotation system ρ 0 (G). Suppose that the edge [u, ] is marked as k + -tisted, ithin an extended rotation system ρ 0(G) that corresponds to a general rotation system obtained from ρ 0 (G) by assigning the tist-type of [u, ] to 1 (if k is odd) or to 0 (if k is even). We change the cyclic eaving structure σ 0 corresponding to this edge-tist and obtain a ne cyclic eaving structure σ 0, as follos. For the to segments 0 u, 0 and 1 u, 1 in σ 0 induced from [u, ], suppose that the tail of 0 u, 0 is connected ith the head of 0, x and that the tail of 1 u, 1 is connected ith the head of 1, y, see Figures 13(A1) and Figure 13(B1). A formal description of the procedure that k-tists the edge [u, ] is given by Algorithm 2, hich has not previously been published. Algorithm 2 Segment-Tist. Segment-Tist([u, ]) Input: an edge [u, ] ith to parallel segments 0 u, 0 and 1 u, 1 in σ 0. Output: the edge [u, ] is k-tisted. 1. Cut the joint of 0 u, 0 and 0, x and the joint of 1 u, 1 and 1, y (as in Figure 13(A2) and Figure 13(B2)); 2. Cross the segments 1 u, 1 and 0 u, 0 k times such that at the first cross point, the segment 1 u, 1 goes above the segment 0 u, 0 (as in Figure 13(A3) and Figure 13(B3)); 3. If k is odd, then rename 0 u, 0 as 0 u, 1, and rename 1 u, 1 as 1 u, 0 (as in Figure 13(A4)). 4. Connect the tail of u, 1 ith the head of 1, y, and connect the tail of u, 0 ith the head of 0, x (as in Figure 13(A5) and Figure 13(B4)). The resulting cyclic eaving σ 0 on the surface S is said to be induced by the extended rotation system ρ 0(G). The eaving structure change corresponding to a k -tist of an edge in the pure rotation system ρ 0 (G) can be described similarly, except for Step 2, here e require after crossing the segments, that the segment 0 u, 0 go above the segment 1 u, 1 at their first crossing point. We may apply the edge-tisting operation on more than one edge in the pure rotation system ρ 0 (G). Figure 14 provides an example to illustrate the above idea. Figure 14(a) is an orientable surface S (i.e., the sphere). Figure 14(b) shos an imbedding π 0 (G) of a graph G on the surface S (the graph edges are given as flat bands). Figure 14(c) is the pure rotation system ρ 0 (G) of G corresponding to the imbedding π 0 (G) in Figure 14(b). Figure 14(e) shos the initial cyclic eaving σ 0 induced from the rotation system ρ 0 (G) in Figure 14(c). Figure 14(g) applies the

23 Extended Graph Rotation Systems as a Model for Cyclic Weaving (a) 23 (b) (c) (d) (e) (f) (g) (h) (j) Figure 14: Cyclic eaving based on rotation systems 1+ -tisting operation on three of the edges in the rotation system ρ0 (G), resulting in an extended rotation system ρ(g). Figure 14(j) is the cyclic eaving σ induced from the extended rotation system ρ(g). We say that an edge is tisted positively if it is k + -tisted for an integer k 1, and that it is tisted negatively if it is k -tisted for an integer k 1. The theorem belo is a foundation for cyclic plain-eaving. Theorem 6.3 Let ρ0 (G) be a pure rotation system for an imbedding π0 : G S of a graph on an orientable surface. Let A be an arbitrary subset of edges of G. If e either tist all edges in A positively or tist all edges in A negatively, then the resulting extended rotation system induces a cyclic plain-eaving on S. Proof. We adopt the same notations as before. Let the cyclic eaving induced by the pure rotation system ρ0 (G) be σ0, here for each face F in π0 (G) there is a corresponding circle cf in σ0 mapped to the surface S, such that no to such circles intersect on S. We provide a detailed proof of the theorem for the case here all edges in the subset A are tisted positively, by induction on the number h of edges in A. The case here all edges in A are tisted negatively can be proved similarly. For the subset A of h edges, let ρh (G) denote the extended rotation system obtained by positively tisting all h edges in A in the given pure rotation system ρ0 (G). We may regard ρ0 (G) as an extended rotation system in hich every edge is 0-tisted. For h 1, e choose an edge [u, ] in A that is k + -tisted in ρh (G) for some integer k > 0, and e let ρh 1 (G) be the extended rotation system obtained by replacing the k + -tisted edge [u, ] in ρh (G) by an untisted edge [u, ]. Thus, ρh 1 can be obtained from ρ0 (G) by positively tisting the h 1 edges in A {[u, ]}, and ρh (G) can be obtained from the rotation system ρh 1 by k + -tisting the edge [u, ]. Let σh and σh 1 be the cyclic eavings induced by the extended rotation systems ρh (G) and ρh 1 (G), respectively. Claim 1. Every segment in the cyclic eaving σh is of the form t1 hv1, v2 it2, here [v1, v2 ] is an edge in the graph G, here t1, t2 {0, 1} are trace-types such that t2 = t1 + t-type([v1, v2 ]) (mod 2), and here t-type([v1, v2 ]) is 1 if [v1, v2 ] is j + -tisted in ρh (G) for an odd number j or is 0 if [v1, v2 ] is untisted or j + -tisted for an even number j. Claim 1 is obviously true for h = 0: all edges in ρ0 (G) are untisted, and all segments in σ0 are of the form 0 hv1, v2 i0 and 1 hv1, v2 i1. By ay of induction, e suppose that for some h > 0, Claim 1 is true for the eaving σh 1. The rotation system ρh (G) is obtained from ρh 1 (G) by k + -tisting the edge [u, ]. By the