Knots and surfaces in 3-dimensional space
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1 March 13, 2012
2 Knots Definition A knot is a smooth embedding of the circle into 3-dimensional space.
3 Knots Figure: A knot
4 Knots Fact I: The mathematical study of knots includes several different branches, each with a very different flavour : algebraic, geometric,...
5 Knots Fact I: The mathematical study of knots includes several different branches, each with a very different flavour : algebraic, geometric,... Fact II: Knots relate to nature, for instance via the tangles in long strands of DNA.
6 Seifert surfaces Definition Given a knot K in 3-dimensional space, an orientable (2-sided) surface S that has boundary K is called a spanning surface.
7 Seifert Surfaces Figure: A Seifert surface Theorem (Seifert) Every knot admits a spanning surface.
8 Seifert surfaces Theorem (Seifert) Every knot admits a spanning surface.
9 Seifert surfaces Theorem (Seifert) Every knot admits a spanning surface. Seifert s algorithm Step 1: Consider a projection of the knot and give it an orientation. Step 2: Resolve each crossing to obtain Seifert circles.
10 Seifert surfaces Theorem (Seifert) Every knot admits a spanning surface. Seifert s algorithm Step 1: Consider a projection of the knot and give it an orientation. Step 2: Resolve each crossing to obtain Seifert circles. Step 3: The Seifert circles bound disks with a natural orientation.
11 Seifert surfaces Theorem (Seifert) Every knot admits a spanning surface. Seifert s algorithm Step 1: Consider a projection of the knot and give it an orientation. Step 2: Resolve each crossing to obtain Seifert circles. Step 3: The Seifert circles bound disks with a natural orientation. Step 4: Connect disks via bands to obtain a spanning surface.
12 Seifert Surfaces For a surface S constructed via Seifert s algorithm, the Euler characteristic of S, denoted by χ(s), can be computed as χ(s) = #disks #bands Definition A Seifert surface for a knot K is a spanning surface of maximal Euler characteristic.
13 Seifert Surfaces (Results of Julian Eisner) Fact I: Many knots K in S 3 admit non-isotopic Seifert surfaces. (Eisner 1977)
14 Seifert Surfaces (Results of Julian Eisner) Fact I: Many knots K in S 3 admit non-isotopic Seifert surfaces. (Eisner 1977) Fact II: Many knots K in S 3 admit disjoint non-isotopic Seifert surfaces. (Eisner 1977)
15 Seifert Surfaces Figure: Connect sum of knots
16 Seifert Surfaces Figure: Connect sum of knots
17 Seifert Surfaces Figure: Schematic
18 Kakimizu complex Definition The vertices of the Kakimizu complex Kak(K) of a knot K in S 3 are given by the isotopy classes of minimal genus Seifert surfaces for K.
19 Kakimizu complex Definition The vertices of the Kakimizu complex Kak(K) of a knot K in S 3 are given by the isotopy classes of minimal genus Seifert surfaces for K. The n-simplices of the Kakimizu complex of K, for n > 1, are given by n-tuples of vertices that admit representatives that are pairwise disjoint.
20 Examples of Kakimizu complexes Example I: Fibered knots have trivial Kakimizu complexes.
21 Examples of Kakimizu complexes Example I: Fibered knots have trivial Kakimizu complexes. Example II: Hyperbolic knots have finite Kakimizu complexes.
22 Structure of Kakimizu complexes Theorem (Scharlemann-Thompson) The Kakimizu complex of a knot is connected. Not stated in these terms.
23 Structure of Kakimizu complexes Theorem (Kakimizu) Suppose that K 1, K 2 are knots with unique minimal genus Seifert surfaces (up to isotopy). Then the Kakimizu complex of K = K 1 #K 2 is a bi-infinite ray. More recently, Kakimizu computed the Kakimizu complexes for all prime knots with up to 10 crossings.
24 Structure of Kakimizu complexes Theorem (Banks) Suppose that K 1, K 2 are knots, then the Kakimizu complex of K 1 #K 2 is the product of three complexes: The Kakimizu complex of K 1, the Kakimizu complex of K 2 and the complex that has underlying space R and vertices at the integers.
25 Structure of Kakimizu complexes Theorem (Banks) There exist knots with locally infinite Kakimizu complex. Theorem (Banks) A knot has locally infinite Kakimizu complex only if it is a satellite of either a torus knot, a cable knot or a connected sum, with winding number 0.
26 Jessica Banks results Figure: An essential torus in a knot complement
27 Structure of the Kakimizu complex Theorem (S 2007) The Kakimizu complex of a knot K is a flag complex.
28 Structure of the Kakimizu complex Theorem (S 2007) The Kakimizu complex of a knot K is a flag complex. Theorem (Kapovich 2009) Let M be a Riemannian 3-manifold with smooth strictly convex boundary, together with a compact family J of smooth curves on M. Let f i : (S i, S i ) (M, J ), i = 1,..., n be incompressible surfaces which are pairwise non-isotopic and pairwise disjoint. Let g i : (S i, S i ) (M, J ), i = 1,..., n be relative area minimizers in the proper isotopy classes of f i, i = 1,..., n. Then g 1 (S 1 ),..., g n (S n ) are also pairwise disjoint.
29 Contractibility of the Kakimizu complex Theorem (Przytycki-S 2010) The Kakimizu complex of a knot is contractible.
30 A projection map on the Kakimizu complex Implicit in Kakimizu s work is a projection map (coming from considerations involving covering spaces and Kakimizu s formulation of the distance on the Kakimizu complex) that, given two vertices v, w, produces a vertex π v (w) that is one step closer to v than w. d(v, π v (w)) = d(v, w) 1
31 Contractibility of the Kakimizu complex Theorem (Przytycki-S 2010) The Kakimizu complex of a knot is contractible. Idea of proof: Choose a vertex v in Kak(K) and prove that the projection map onto v is a contraction of Kak(K). Challenge: Make sure that the projection map behaves well on links of vertices.
32 Connectedness of the Kakimizu complex Theorem (Scharlemann-Thompson) The Kakimizu complex of a knot is connected. Idea of new proof: Given any two vertices v, w, we construct a path
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