The combinatorics of CAT(0) cube complexes

Transcription

1 The combinatorics of CAT() cube complexes (and moving discrete robots efficiently) Federico Ardila M. San Francisco State University, San Francisco, California. Universidad de Los Andes, Bogotá, Colombia. MIT Combinatorics Seminar November 4, 22

2 motivation 8 preliminaries examples characterizations applications Joint work with: Megan Owen (UC Berkeley), Seth& V. Sullivant R. GHRIST PETERSON(NCSU) Diego Cifuentes (Los Andes) Rika Yatchak (SFSU) Tia Baker (SFSU) F IGURE 4. A positive articulated robot arm example [left] with fixed endpoint. One generator [center] flips corners and has as its trace the central four edges. The other generator [right] rotates the end of. There are many CAT() cube complexes in nature". the arm, and has trace equal to the two activated edges. The two points I would like to make. 2. CAT() cube complexes have an elegant and useful structure.

3 . MOTIVATION. Moving robots. A robotic snake can move:. the head or tail or 2. a joint without self-intersecting. Snake: Moves: : 2:

4 Motivation: moving robots. How do we move the robotic snake (optimally) using these moves from one position to another one? Position Position 2

5 Motivation: an easier question. How am I going to get to the Dartmouth tonight? MIT Dartmouth

6 Motivation: an easier question. How am I going to get to the Dartmouth tonight, optimally? With a map! Let s do the same! Let s build a map of all possible positions of the robot.

7 Motivation: back to moving robots. Let s build a map of all possible positions of the robot. A small piece: (discrete model)

8 Motivation: moving robots. Let s build a map of all possible positions of the robot. A small piece: (continuous model)

9 Motivation: moving robots. Let s build a map of all possible positions. A complete example: A CAT() cube complex!! How can we understand them? Navigate them?

10 Motivation: moving robots. RACG vs. RAMRG RACG: RAMRG: How can we understand CAT() cube complexes? How should we navigate them? Obstacles: High dimension. Complicated ramification. This is what we need to overcome.

11 The plan:. (One) motivation: moving robots. 2. Preliminaries: CAT() spaces and cube complexes. 3. There are many CAT() cube complexes in nature". (Three examples.) 4. CAT() cube complexes have an elegant structure. (Two characterizations, two examples.) 5. This structure is useful. (Three applications.)

12 2. PRELIMINARIES. CAT() spaces A metric space X is CAT() if it has non-positive curvature everywhere, in the sense that triangles in X are thinner" than flat triangles. Roughly, it is saddle shaped". More precisely, we require: There is a unique geodesic path between any two points of X. (CAT() inequality) Consider any triangle T in X and a comparison triangle T of the same sidelengths in the Euclidean plane R 2. Consider any chord (of length d) in T and the corresponding chord (of length d ) in T. Then d d. X R 2 a b d c a d c b

13 PRELIMINARIES. Cube complexes A cube complex is a space obtained by gluing cubes (of possibly different dimensions) along their faces. (Like a simplicial complex, but the building blocks are cubes.) Metric: Euclidean inside each cube. We are interested in cube complexes which are CAT().

14 Example. Five squares glued around a corner.

15 Example. Five squares glued around a corner. A' A C B B' C' AB = AC = 5 A B = A C = 5 BC = 2 2 B C = 2 2

16 Example. Five squares glued around a corner. A' A E C D' E' D B B' C' AB = AC = 5 A B = A C = 5 BC = 2 2 B C = 2 2 DE = < 2 = D E. This triangle is thin. (But: I still need to test many chords.) This space is CAT(). (But: I still need to test many triangles.) Not so practical, huh?

17 3. EXAMPLES. Example. Robot motion planning State complex. vertices = positions. edges = moves. cubes = physically independent" moves. Theorem (GP) This is often a CAT() cube complex. This works very generally for many reconfiguration systems, where we change vertex labels on a graph using local moves.

18 Example 2. Phylogenetic trees (Billera, Holmes, Vogtmann): Goal: Predict the evolutionary tree of n current-day species/languages/... Approach: Build a space T n of all possible trees. Study it, navigate it b a c b a c Thm. (BHV) T n is a CAT() cube complex. Cor. T n has unique geodesics. Cor. Average" trees exist.

19 Example 3. Geometric Group Theory. A right-angled Coxeter group is a group of the form Example W (G) = v V v 2 Regular = for v V, (uv) 2 CAT() Cube = for uv E Example: a 2 = Example b 2 = c 2 = d 2 = Complexes Rick Scott Example For the graph a Intro to cube complexes Cube complexes CAT() (ab) 2 = (ac) 2 = (bc) 2 = (cd) 2 = Vertex-regular b Regular CAT() Cube Key Examples Complexes RACG s Example RAMRG s Thm. (Davis) Rick Scott Right-angled Coxeter groups are CAT(): bc For the graph the Davis complexgrowth is series a Definition W (G) acts Intro to cube very nicely" on a CAT() cube Propertiescomplex of CAT() c d cube complexes abc X(G). complexes ac Cube complexes Recurrence relations CAT() Growth formula Vertex-regular Examples b Remarks Key Examples Proof sketch b RACG s RAMRG s Growth series Definition Properties of CAT() cube complexes Recurrence relations Growth formula Examples Remarks Proof sketch abc b bc ac c cd d ab a c the Davis d c cd d ab a GGT uses the geometry of X(G) to study the group W (G); e.g., If a group G is CAT(), the word problem" is easy for G.

20 Regular CAT() Cube Complexes Rick Scott Intro to cube complexes Cube complexes CAT() Vertex-regular Key Examples RACG s RAMRG s Growth series Definition Properties of CAT() cube complexes Recurrence relations Growth formula Examples 4. CHARACTERIZATIONS. Which cube complexes are CAT()? Vertex Links In general, CAT() is a subtle condition; but for cube complexes:. Gromov s characterization. Theorem. In a cubical (Gromov, complex, 987) the links A cube of vertices complex are is simplicial CAT() if complexes. and only if it is simply connected and the link of every vertex is a flag simplicial complex. A simplicial complex L is a flag complex if whenever the flag: -skeleton if the -skeleton of a simplex of a simplex occurs T is in, then T is in. in L, so does the entire simplex.

21 Characterizations: Which cube complexes are CAT()? 2. Our characterization. RACG vs. RAMRG Regular CAT() Cube Complexes RACG: Rick Scott Intro to cube complexes Cube complexes CAT() Vertex-regular Key Examples RACG s RAMRG s Growth series RAMRG: Definition Properties of CAT() cube complexes Recurrence relations Growth formula Examples Remarks Theorem. (A-Owen-Sullivant 8) Proof sketch (Pointed) CAT() cube complexes are in bijection with posets with inconsistent pairs PIP: A poset P and a set of inconsistent pairs" {x, y}, with x, y inconsistent, y < z x, z inconsistent.

22 Theorem. (A-Owen-Sullivant 8) (Pointed) CAT() cube complexes are in bijection with posets with inconsistent pairs. Sketch of proof. Idea: CAT() cube complexes look like" distributive lattices. So imitate Birkhoff s bijection: distributive lattices posets : X has hyperplanes which split cubes in half. (Sageev)

23 Theorem. (A. - Owen - Sullivant 8) (Pointed) CAT() cube complexes are in bijection with posets with inconsistent pairs. Bijection. : Fix a home" vertex v v If i, j are hyperplanes, declare: i < j if one needs to cross i before crossing j i, j inconsistent if it is impossible to cross them both.

24 Remark. Sageev (95) and Roller (98) obtained a different combinatorial description. Which one is more useful depends on the context. Let s see some applications.

25 Example. The pinned-down robotic snake in a square grid. Theorem. (A.-Baker-Yatchak, 22) The state complex is a CAT() cubical complex. Its PIP ( remote control") is as shown. Complex of 2 n states in n 2 dim. 2 n2 buttons".

26 Example. The pinned-down robotic snake in a square grid. Corollary. (A.-Baker-Yatchak, 22) Let q n,d be the number of d-cubes in the state complex for the robotic arm of length n. Then q n,d x n y d + xy = 2x x 2 y. n,d

27 Example 2. The pinned-down robotic snake in a strip. Theorem. (A.-Baker-Yatchak, 22) The state complex is a CAT() cubical complex. Its PIP ( remote control") is as shown. Complex of F n ( ) n states in n n2 3-dim 4 buttons.

28 Example 2. The pinned-down robotic snake in a strip Corollary. (A.-Baker-Yatchak, 22) Let s n,d be the number of d-cubes in the state complex for the robotic arm of length n. Then s n,d x n y d = + x + xy + x 2 y x x 2 x 3 y. n,d

29 Example 3. The non-pinned-down robotic snake. A negative result: Theorem. (A. - Yatchak, 22) If the snake in a grid is not pinned down, the state complex is not always CAT(). Open question. Which robots give CAT() cube complexes?

30 Application. Embeddability conjecture. Conjecture. (Niblo, Sageev, Wise) Any finite d-dimensional CAT() cube complex can be embedded in the cubing Z d v Proof. (AOS 8) Dilworth already showed (in 95!) how to embed J(Q) in Z d : Write Q as a union of d disjoint chains. (Example: 246, 35, ) Straighten" the cube complex along each chain. (Proof also by Brodzki, Campbell, Guentner, Niblo, Wright (8).)

31 Application. Embeddability conjecture. Conjecture. (Niblo, Sageev, Wise) Any finite d-dimensional CAT() cube complex can be embedded in the cubing Z d v Proof. (AOS 8) Dilworth already showed (in 95!) how to embed J(Q) in Z d : Write Q as a union of d disjoint chains. (Example: 246, 35, ) Straighten" the cube complex along each chain. (Proof also by Brodzki, Campbell, Guentner, Niblo, Wright (8).)

32 Application 2. All CAT() cube complexes are robotic". Theorem. (Ghrist-Peterson 7) Every CAT() cube complex can be realized as a state complex. Their proof is indirect v Alternative proof. (AOS ) Root X poset with inconsistent pairs P. A virus robot" takes over the poset P. It can take over a new cell q if and only if: o it already took over all elements p < q, and o it hasn t taken over any elements inconsistent with q. Then X is the state complex for this robot.

33 Application 2. All CAT() cube complexes are robotic". Theorem. (Ghrist-Peterson 7) Every CAT() cube complex can be realized as a state complex. Their proof is indirect v Alternative proof. (AOS ) Root X poset with inconsistent pairs P. A virus robot" takes over the poset P. It can take over a new cell q if and only if: o it already took over all elements p < q, and o it hasn t taken over any elements inconsistent with q. Then X is the state complex for this robot.

34 Application 3. The Hopf algebra of CAT() cube complexes. Theorem. (A. - Cifuentes 2) CAT() cube complexes have the structure of a Hopf algebra. There is an elegant formula for the antipode.

35 Application 4. Finding geodesics in CAT() cube complexes. Motivation: Algorithm. (Owen-Provan 9) A polynomial-time algorithm to find the geodesic between trees T and T 2 in the space of trees T n. ( 2-approx.: Amenta 7, exp.: GeoMeTree 8, GeodeMaps 9) This allows us to find distances between trees average" trees.

36 Application 4. Finding geodesics in CAT() cube complexes. We use combinatorial description ( remote control") of X to get: Algorithm. (AOS, ) An algorithm to find the geodesic between points p and q in any CAT() cube complex X. This allows us to navigate the state complex of any reconfiguration system find the optimal robot motion between two positions. (Polynomial time? Computer implementation?)

37 many thanks The paper and slides are available at: Advances in Applied Mathematics 48 (22)