Jones polynomials and incompressible surfaces

Transcription

1 Jones polynomials and incompressible surfaces joint with D. Futer and J. Purcell Geometric Topology in Cortona (in honor of Riccardo Benedetti for his 60th birthday), Cortona, Italy, June 3-7, 2013 David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

2 Given: Diagram of a knot or link Quantum Topology Knot invariants invariants esp. colored Jones polynomials Talk Goal: Discuss a setting where, under certain knot diagrammatic hypothesis, we study both sides and derive relations between them. Geometric topology Incompressible surfaces in knot complements Geometric structures and data esp. hyperbolic geometry and volume David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

3 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

4 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

5 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

6 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

7 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

8 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) Colored Jones polynomial (CPJ) relations: Boundary slopes relate to degrees of CJP. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

9 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) Colored Jones polynomial (CPJ) relations: Boundary slopes relate to degrees of CJP. Coefficients measure how far surfaces are from being fibers David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

10 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) Colored Jones polynomial (CPJ) relations: Boundary slopes relate to degrees of CJP. Coefficients measure how far surfaces are from being fibers detect geometric types of surfaces David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

11 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) Colored Jones polynomial (CPJ) relations: Boundary slopes relate to degrees of CJP. Coefficients measure how far surfaces are from being fibers detect geometric types of surfaces Guts relate CJP to volume of hyperbolic knots. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

12 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) Colored Jones polynomial (CPJ) relations: Boundary slopes relate to degrees of CJP. Coefficients measure how far surfaces are from being fibers detect geometric types of surfaces Guts relate CJP to volume of hyperbolic knots. Method-Tools: Create ideal polyhedral decomposition of surface complements. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

13 Outline Setting: Given knot diagram construct state graphs (ribbon graphs).. Build state surfaces spanned by the knot... Ribbon graphs relate to Jones polynomials... Give diagrammatic conditions for state surface incompressibility. Understand JSJ-decompositions of surface complements... emphasis on hyperbolic part ( the Guts ) Colored Jones polynomial (CPJ) relations: Boundary slopes relate to degrees of CJP. Coefficients measure how far surfaces are from being fibers detect geometric types of surfaces Guts relate CJP to volume of hyperbolic knots. Method-Tools: Create ideal polyhedral decomposition of surface complements. Use normal surface theory to get correspondence topology of surface complement state graph topology David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

14 State Graphs Two choices for each crossing, A or B resolution. Choice of A or B resolutions for all crossings: state σ. Result: Planar link without crossings. Components: state circles. Form a graph by adding edges at resolved crossings. Call this graph H σ. ( Note: n crossings 2 n state graphs) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

15 Example states Link diagram All A state All B state Above: H A and H B. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

16 Example states Link diagram All A state All B state Above: H A and H B. The Jones polynomial of the knot can be calculated from H A or H B : spanning graph expansion arising from the Bollobas-Riordan ribbon graph polynomial (Turaev, Dasbach-Futer-K-Lin-Stoltzfus). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

17 Colored Jones polynomial prelims For a knot K, and n = 1, 2,..., we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β nt kn+1 + α nt kn. Some properties: J K,n (t) is determined by the Jones polynomials of certain cables of K. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

18 Colored Jones polynomial prelims For a knot K, and n = 1, 2,..., we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β nt kn+1 + α nt kn. Some properties: J K,n (t) is determined by the Jones polynomials of certain cables of K. The sequence {J K,n (t)} n is q-holonomic: for every knot the CJP s satisfy linear recursion relations in n (Garoufalidis-Le, 2004). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

19 Colored Jones polynomial prelims For a knot K, and n = 1, 2,..., we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β nt kn+1 + α nt kn. Some properties: J K,n (t) is determined by the Jones polynomials of certain cables of K. The sequence {J K,n (t)} n is q-holonomic: for every knot the CJP s satisfy linear recursion relations in n (Garoufalidis-Le, 2004). Then, for every K, Degrees m n, k n are quadratic (quasi)-polynomials in n. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

20 Colored Jones polynomial prelims For a knot K, and n = 1, 2,..., we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β nt kn+1 + α nt kn. Some properties: J K,n (t) is determined by the Jones polynomials of certain cables of K. The sequence {J K,n (t)} n is q-holonomic: for every knot the CJP s satisfy linear recursion relations in n (Garoufalidis-Le, 2004). Then, for every K, Degrees m n, k n are quadratic (quasi)-polynomials in n. Coefficients α n, β n... satisfy recursive relations in n. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

21 Colored Jones polynomial prelims For a knot K, and n = 1, 2,..., we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β nt kn+1 + α nt kn. Some properties: J K,n (t) is determined by the Jones polynomials of certain cables of K. The sequence {J K,n (t)} n is q-holonomic: for every knot the CJP s satisfy linear recursion relations in n (Garoufalidis-Le, 2004). Then, for every K, Degrees m n, k n are quadratic (quasi)-polynomials in n. Coefficients α n, β n... satisfy recursive relations in n. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

22 Colored Jones polynomial prelims For a knot K, and n = 1, 2,..., we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β nt kn+1 + α nt kn. Some properties: J K,n (t) is determined by the Jones polynomials of certain cables of K. The sequence {J K,n (t)} n is q-holonomic: for every knot the CJP s satisfy linear recursion relations in n (Garoufalidis-Le, 2004). Then, for every K, Degrees m n, k n are quadratic (quasi)-polynomials in n. Coefficients α n, β n... satisfy recursive relations in n. Remark. Properties manifest themselves in strong forms for knots with state graphs that have no edge with both endpoints on a single state circle -That is when K is A-adequate (next) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

23 Semi-adequate links Lickorish Thistlethwaite 1987: Introduced A adequate links (B adequate links) in the context of Jones polynomials. Definition A link is A adequate if has a diagram with its graph H A has no edge with both endpoints on the same state circle. Similarly B-adequate. Semi-adequate: A or B-adequate. Some examples: David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

24 Semi-adequate links are abundant! Some familiar classes and their geometry: all but two of prime knots up to 11 crossings. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

25 Semi-adequate links are abundant! Some familiar classes and their geometry: all but two of prime knots up to 11 crossings. all alternating knots, (prime are torus links or hyperbolic), David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

26 Semi-adequate links are abundant! Some familiar classes and their geometry: all but two of prime knots up to 11 crossings. all alternating knots, (prime are torus links or hyperbolic), all Montesinos knots (mostly hyperbolic), David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

27 Semi-adequate links are abundant! Some familiar classes and their geometry: all but two of prime knots up to 11 crossings. all alternating knots, (prime are torus links or hyperbolic), all Montesinos knots (mostly hyperbolic), all positive (negative) knots (lots of hyperbolic), David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

28 Semi-adequate links are abundant! Some familiar classes and their geometry: all but two of prime knots up to 11 crossings. all alternating knots, (prime are torus links or hyperbolic), all Montesinos knots (mostly hyperbolic), all positive (negative) knots (lots of hyperbolic), many arborescent knots (mostly hyperbolic), all closed 3-braids (prime are torus knots or hyperbolic (Stoimenow), large families of hyperbolic braid and plat closures (A. Giambrone), blackboard cables and Whitehead doubles of semi-adequate knots (satellites) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

29 Semi-adequate links are abundant! Some familiar classes and their geometry: all but two of prime knots up to 11 crossings. all alternating knots, (prime are torus links or hyperbolic), all Montesinos knots (mostly hyperbolic), all positive (negative) knots (lots of hyperbolic), many arborescent knots (mostly hyperbolic), all closed 3-braids (prime are torus knots or hyperbolic (Stoimenow), large families of hyperbolic braid and plat closures (A. Giambrone), blackboard cables and Whitehead doubles of semi-adequate knots (satellites) Question: Is there an algorithm to decide whether a given knot is semi-adequate? David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

30 CJP of semi-adequate links Collapse each state circle of H A to a vertex to obtain the state graph G A. Remove redundant edges to obtain the reduced state graph G A. J K,n (t) := α n t mn + β n t mn β n tkn+1 + α n tkn. Extreme Coefficients stabilize; they depend only on G A! David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

31 CJP of semi-adequate links Collapse each state circle of H A to a vertex to obtain the state graph G A. Remove redundant edges to obtain the reduced state graph G A. J K,n (t) := α n t mn + β n t mn β n tkn+1 + α n tkn. Extreme Coefficients stabilize; they depend only on G A! (Lickorish-Thistlethwaite) α n = 1; independent of n David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

32 CJP of semi-adequate links Collapse each state circle of H A to a vertex to obtain the state graph G A. Remove redundant edges to obtain the reduced state graph G A. J K,n (t) := α n t mn + β n t mn β n tkn+1 + α n tkn. Extreme Coefficients stabilize; they depend only on G A! (Lickorish-Thistlethwaite) α n = 1; independent of n (Dasbach-Lin/ Stoimenow) β K := β n = 1 χ(g A ), n > 1. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

33 CJP of semi-adequate links Collapse each state circle of H A to a vertex to obtain the state graph G A. Remove redundant edges to obtain the reduced state graph G A. J K,n (t) := α n t mn + β n t mn β n tkn+1 + α n tkn. Extreme Coefficients stabilize; they depend only on G A! (Lickorish-Thistlethwaite) α n = 1; independent of n (Dasbach-Lin/ Stoimenow) β K := β n = 1 χ(g A ), n > 1. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

34 CJP for semi-adequate links, con t ( Armond) (the abs. values of) m-th to last coefficients of J K,n (t) is independent on n, for n m. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

35 CJP for semi-adequate links, con t ( Armond) (the abs. values of) m-th to last coefficients of J K,n (t) is independent on n, for n m.the Tail of the CJP T K (t) = βk m tm = 1 + β K t + O(t2 ), βk m =m-th to last coefficients stabilized coefficient. m David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

36 CJP for semi-adequate links, con t ( Armond) (the abs. values of) m-th to last coefficients of J K,n (t) is independent on n, for n m.the Tail of the CJP T K (t) = βk m tm = 1 + β K t + O(t2 ), βk m =m-th to last coefficients stabilized coefficient. m (Armond-Dasbach) T K (t) only depends on reduced state graph, G A David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

37 CJP for semi-adequate links, con t ( Armond) (the abs. values of) m-th to last coefficients of J K,n (t) is independent on n, for n m.the Tail of the CJP T K (t) = βk m tm = 1 + β K t + O(t2 ), βk m =m-th to last coefficients stabilized coefficient. m (Armond-Dasbach) T K (t) only depends on reduced state graph, G A (Garoufalidis-Le) Discovered higher order stability phenomena in CJP ( higher order tails ); gave closed formulae for the tails. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

38 CJP for semi-adequate links, con t ( Armond) (the abs. values of) m-th to last coefficients of J K,n (t) is independent on n, for n m.the Tail of the CJP T K (t) = βk m tm = 1 + β K t + O(t2 ), βk m =m-th to last coefficients stabilized coefficient. m (Armond-Dasbach) T K (t) only depends on reduced state graph, G A (Garoufalidis-Le) Discovered higher order stability phenomena in CJP ( higher order tails ); gave closed formulae for the tails. Extreme degrees of CJP (Thistlethwaite) D any diagram of K, c (D)=number of negative crossings in D. Then k n n 2 2c (D) + O(n), k n :=min deg J K,n (t). We have equality exactly when D is A-adequate. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

39 CJP for semi-adequate links, con t ( Armond) (the abs. values of) m-th to last coefficients of J K,n (t) is independent on n, for n m.the Tail of the CJP T K (t) = βk m tm = 1 + β K t + O(t2 ), βk m =m-th to last coefficients stabilized coefficient. m (Armond-Dasbach) T K (t) only depends on reduced state graph, G A (Garoufalidis-Le) Discovered higher order stability phenomena in CJP ( higher order tails ); gave closed formulae for the tails. Extreme degrees of CJP (Thistlethwaite) D any diagram of K, c (D)=number of negative crossings in D. Then k n n 2 2c (D) + O(n), k n :=min deg J K,n (t). We have equality exactly when D is A-adequate. Thus k n is a quadratic polynomial in n; can be calculated explicitly. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

40 CJP for semi-adequate links, con t ( Armond) (the abs. values of) m-th to last coefficients of J K,n (t) is independent on n, for n m.the Tail of the CJP T K (t) = βk m tm = 1 + β K t + O(t2 ), βk m =m-th to last coefficients stabilized coefficient. m (Armond-Dasbach) T K (t) only depends on reduced state graph, G A (Garoufalidis-Le) Discovered higher order stability phenomena in CJP ( higher order tails ); gave closed formulae for the tails. Extreme degrees of CJP (Thistlethwaite) D any diagram of K, c (D)=number of negative crossings in D. Then k n n 2 2c (D) + O(n), k n :=min deg J K,n (t). We have equality exactly when D is A-adequate. Thus k n is a quadratic polynomial in n; can be calculated explicitly. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

41 State surface Given a state σ, using graph H σ and link diagram, form the state surface S σ. Each state circle bounds a disk in S σ (nested disks drawn on top). At each edge (for each crossing) attach twisted band. A resolution B resolution David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

42 Example state surfaces Fig-8 knot S A S B Seifert surface For alternating knots: S A and S B are checkerboard surfaces. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

43 The surface S A and CJP: Boundary slopes Theorem (Ozawa, FKP) The surface S A = S A (D) is essential in S 3 K D(K) is A adequate. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

44 The surface S A and CJP: Boundary slopes Theorem (Ozawa, FKP) The surface S A = S A (D) is essential in S 3 K D(K) is A adequate. Ozawa s proof was first We give information about the topology of the surface complement, S 3 \\S A, in terms of colored Jones polynomials. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

45 The surface S A and CJP: Boundary slopes Theorem (Ozawa, FKP) The surface S A = S A (D) is essential in S 3 K D(K) is A adequate. Ozawa s proof was first We give information about the topology of the surface complement, S 3 \\S A, in terms of colored Jones polynomials. S A = state surface with K = S A an A-adequate knot (one component). The class [K] in H 1 ( (S 3 K)) is determined by an element in Q { }, called the boundary slope of S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

46 The surface S A and CJP: Boundary slopes Theorem (Ozawa, FKP) The surface S A = S A (D) is essential in S 3 K D(K) is A adequate. Ozawa s proof was first We give information about the topology of the surface complement, S 3 \\S A, in terms of colored Jones polynomials. S A = state surface with K = S A an A-adequate knot (one component). The class [K] in H 1 ( (S 3 K)) is determined by an element in Q { }, called the boundary slope of S A. Theorem (FKP) For an A adequate diagram, 4 bdry slope of S A = lim n n 2 k n, k n :=min deg J K,n (t). There is a similar statement for B-adequate knots. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

47 The surface S A and CJP: Boundary slopes Theorem (Ozawa, FKP) The surface S A = S A (D) is essential in S 3 K D(K) is A adequate. Ozawa s proof was first We give information about the topology of the surface complement, S 3 \\S A, in terms of colored Jones polynomials. S A = state surface with K = S A an A-adequate knot (one component). The class [K] in H 1 ( (S 3 K)) is determined by an element in Q { }, called the boundary slope of S A. Theorem (FKP) For an A adequate diagram, 4 bdry slope of S A = lim n n 2 k n, k n :=min deg J K,n (t). There is a similar statement for B-adequate knots. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

48 The Jones Slopes Conjecture (Curtis-Taylor) Related -slopes of checkerboard surfaces of alternating knots to degree of Jones polynomial. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

49 The Jones Slopes Conjecture (Curtis-Taylor) Related -slopes of checkerboard surfaces of alternating knots to degree of Jones polynomial. Slopes Conjecture. (Garoufalidis ) For every knot K the sequence { 4 n 2 k n} n, has finitely many cluster points, each of which is a -slope of K. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

50 The Jones Slopes Conjecture (Curtis-Taylor) Related -slopes of checkerboard surfaces of alternating knots to degree of Jones polynomial. Slopes Conjecture. (Garoufalidis ) For every knot K the sequence { 4 n 2 k n} n, has finitely many cluster points, each of which is a -slope of K. Similarly, for m n :=max deg J K,n (t), the sequence { 4 n 2 m n} n, has finitely many cluster points, each of which is a -slope of K. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

51 The Jones Slopes Conjecture (Curtis-Taylor) Related -slopes of checkerboard surfaces of alternating knots to degree of Jones polynomial. Slopes Conjecture. (Garoufalidis ) For every knot K the sequence { 4 n 2 k n} n, has finitely many cluster points, each of which is a -slope of K. Similarly, for m n :=max deg J K,n (t), the sequence { 4 n 2 m n} n, has finitely many cluster points, each of which is a -slope of K. Remarks: q-holonomicity implies that the sets of cluster points above are finite. (Hatcher) Every knot has finitely many -slopes. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

52 What s known For knots that are A and B-adequate slopes conjecture is know for both sides. (Garoufalidis) torus knots, certain 3-string pretzel knots P( 2, p, q) ( A-adequate not B-adequate) For pretzel knots the boundary slopes are all known./ For torus knots CJP has been calculated. (Dunfield Garoufalidis) Verified conjecture for the class of 2-fusion knots. (normal surface theory+character variety techniques to get the incompressible surface). (van der Veen) Formulated a Slopes conjecture for the multi-colored CP of links. Showed that S A verifies it A-adequate links. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

53 The surface S A and CJP: Coefficients For an A-adequate link, β K is the stabilized penultimate coefficient of CJP. Theorem (Futer K Purcell) For an A adequate diagram D(K), the following are equivalent: 1 The penultimate coefficient is β K = 0. 2 The reduced graph G A is a tree. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

54 The surface S A and CJP: Coefficients For an A-adequate link, β K is the stabilized penultimate coefficient of CJP. Theorem (Futer K Purcell) For an A adequate diagram D(K), the following are equivalent: 1 The penultimate coefficient is β K = 0. 2 The reduced graph G A is a tree. 3 S A is a fiber in S 3 K : David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

55 The surface S A and CJP: Coefficients For an A-adequate link, β K is the stabilized penultimate coefficient of CJP. Theorem (Futer K Purcell) For an A adequate diagram D(K), the following are equivalent: 1 The penultimate coefficient is β K = 0. 2 The reduced graph G A is a tree. 3 S A is a fiber in S 3 K : S 3 \\S A is S A I. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

56 The surface S A and CJP: Coefficients For an A-adequate link, β K is the stabilized penultimate coefficient of CJP. Theorem (Futer K Purcell) For an A adequate diagram D(K), the following are equivalent: 1 The penultimate coefficient is β K = 0. 2 The reduced graph G A is a tree. 3 S A is a fiber in S 3 K : S 3 \\S A is S A I. Exercise. Derive Stalling s result: Positive closed braids are fibered with fiber obtained from Seifert s algorithm to the braid diagram. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

57 The surface S A and CJP: Coefficients For an A-adequate link, β K is the stabilized penultimate coefficient of CJP. Theorem (Futer K Purcell) For an A adequate diagram D(K), the following are equivalent: 1 The penultimate coefficient is β K = 0. 2 The reduced graph G A is a tree. 3 S A is a fiber in S 3 K : S 3 \\S A is S A I. Exercise. Derive Stalling s result: Positive closed braids are fibered with fiber obtained from Seifert s algorithm to the braid diagram. Stronger statements: (For a hyperbolic link K ) S A is quasifuchsian iff β K 0 David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

58 The surface S A and CJP: Coefficients For an A-adequate link, β K is the stabilized penultimate coefficient of CJP. Theorem (Futer K Purcell) For an A adequate diagram D(K), the following are equivalent: 1 The penultimate coefficient is β K = 0. 2 The reduced graph G A is a tree. 3 S A is a fiber in S 3 K : S 3 \\S A is S A I. Exercise. Derive Stalling s result: Positive closed braids are fibered with fiber obtained from Seifert s algorithm to the braid diagram. Stronger statements: (For a hyperbolic link K ) S A is quasifuchsian iff β K 0 when β K is large, S A should be far from being a fiber (next). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

59 Is there more in β K? How about in the whole tail? In general, β K measures the size (in the sense of Guts) of the hyperbolic part in Jaco-Shalen-Johannson decomposition S A. This, combined with work of Agol- W. Thurston- Storm gives: large β K implies large volume for S 3 K. What about the tail? Recall T K (t) = 1 + β K t + O(t2 ). Theorem (Armond-Dasbach) Suppose K A-adequate. Then, T K (t) = 1 if and only if β K = 0. Note: if β K = 0 then G A is a tree Thus, T K (t) = 1 if and only if S A is a fiber in S 3 K. Question. If T K (t) 1 does it contain more information about the complement of S A and the geometry of K than β K? David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

60 Topology of the state surface complement M A = S 3 \\S A is obtained by removing a neighborhood of S A from S 3. On M A we have the parabolic locus P = remains from (S 3 K) after cutting along S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

61 Topology of the state surface complement M A = S 3 \\S A is obtained by removing a neighborhood of S A from S 3. On M A we have the parabolic locus P = remains from (S 3 K) after cutting along S A. The annulus version of the JSJ decomposition for the pair (M A, P) assures that M A can be cut along along essential annuli, to obtain three kinds of pieces: 1 I bundles ( e.g. Σ I for Σ S A, although Σ I can also occur), Σ 1 2 Seifert fibered solid tori, 3 Guts(S 3 K, S A ). Thurston showed that the guts admit a hyperbolic metric. Σ 0 David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

62 Topology of Guts and Volume Guts serve as an indication that a surface S A is far from being a fiber. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

63 Topology of Guts and Volume Guts serve as an indication that a surface S A is far from being a fiber. 1 If S A is a fiber of M A = = S A I: no guts. (β K =0) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

64 Topology of Guts and Volume Guts serve as an indication that a surface S A is far from being a fiber. 1 If S A is a fiber of M A = = S A I: no guts. (β K =0) 2 Guts(S 3 K, S A ) = iff M A union of I-bundles and solid tori (book of I-bundles) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

65 Topology of Guts and Volume Guts serve as an indication that a surface S A is far from being a fiber. 1 If S A is a fiber of M A = = S A I: no guts. (β K =0) 2 Guts(S 3 K, S A ) = iff M A union of I-bundles and solid tori (book of I-bundles) 3 If K is a hyperbolic A-adequate link, the guts of a surface S A also have implications for hyperbolic volume via the following theorem: David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

66 Topology of Guts and Volume Guts serve as an indication that a surface S A is far from being a fiber. 1 If S A is a fiber of M A = = S A I: no guts. (β K =0) 2 Guts(S 3 K, S A ) = iff M A union of I-bundles and solid tori (book of I-bundles) 3 If K is a hyperbolic A-adequate link, the guts of a surface S A also have implications for hyperbolic volume via the following theorem: Theorem (Agol Storm Thurston) Let M be a compact 3 manifold with hyperbolic interior of finite volume, and S M an embedded essential surface. Then Vol (M) v 8 χ(guts(m, S)), where v is the volume of a regular ideal octahedron. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

67 The meaning of β K : Special case D(K) =an A-adequate diagram with S A the corresponding all-a state surface. Theorem (F Kalfagianni Purcell) Let D(K) be an A adequate diagram such that every 2 edge loop in G A comes from a twist region. Then the surface S A satisfies χ(guts (S 3 K, S A )) = χ(g A ) = 1 β K twist region David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

68 The meaning of β K : Special case D(K) =an A-adequate diagram with S A the corresponding all-a state surface. Theorem (F Kalfagianni Purcell) Let D(K) be an A adequate diagram such that every 2 edge loop in G A comes from a twist region. Then the surface S A satisfies χ(guts (S 3 K, S A )) = χ(g A ) = 1 β K twist region Corollary Under the same hypotheses, if K is hyperbolic, Vol (S 3 K) v 8 (β K 1). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

69 The meaning of β K : Special case D(K) =an A-adequate diagram with S A the corresponding all-a state surface. Theorem (F Kalfagianni Purcell) Let D(K) be an A adequate diagram such that every 2 edge loop in G A comes from a twist region. Then the surface S A satisfies χ(guts (S 3 K, S A )) = χ(g A ) = 1 β K twist region Corollary Under the same hypotheses, if K is hyperbolic, Vol (S 3 K) v 8 (β K 1). For alternating knots and links, this follows from work of Lackenby and Dasbach Lin. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

70 The meaning of β K : Special case D(K) =an A-adequate diagram with S A the corresponding all-a state surface. Theorem (F Kalfagianni Purcell) Let D(K) be an A adequate diagram such that every 2 edge loop in G A comes from a twist region. Then the surface S A satisfies χ(guts (S 3 K, S A )) = χ(g A ) = 1 β K twist region Corollary Under the same hypotheses, if K is hyperbolic, Vol (S 3 K) v 8 (β K 1). For alternating knots and links, this follows from work of Lackenby and Dasbach Lin. There are large families non-alternating knots satisfying the hypothesis (A. Giambrone) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

71 A worked example D(K) all A state G A = G A David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

72 A worked example D(K) all A state G A = G A 1 β = χ(g A ) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

73 A worked example D(K) all A state G A = G A 1 β = χ(g A ) = χ(s A ) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

74 A worked example D(K) all A state G A = G A 1 β = χ(g A ) = χ(s A ) = χ(s 3 \\S A ) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

75 A worked example D(K) all A state G A = G A 1 β = χ(g A ) = χ(s A ) = χ(s 3 \\S A ) = χ(guts ) David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

76 A worked example D(K) all A state G A = G A 1 β = χ(g A ) = χ(s A ) = χ(s 3 \\S A ) = χ(guts ) = 3 David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

77 A worked example D(K) all A state G A = G A 1 β = χ(g A ) = χ(s A ) = χ(s 3 \\S A ) = χ(guts ) = 3 v 8 ( β 1) = v 8 χ(g A ) = David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

78 A worked example D(K) all A state G A = G A 1 β = χ(g A ) = χ(s A ) = χ(s 3 \\S A ) = χ(guts ) = 3 v 8 ( β 1) = v 8 χ(g A ) = Vol (S 3 K) = David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

79 Sample family: positive braids σ 4 2 σ3 1 σ3 3 σ3 2 σ4 3 David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

80 Sample family: positive braids σ 4 2 σ3 1 σ3 3 σ3 2 σ4 3 Theorem (F Kalfagianni Purcell) Suppose that K is the closure of a positive braid b = σ r 1 i1 σ r 2 i2 σ r k i k, where r j 3 for all j. In other words, there are k twist regions, each with at least 3 crossings. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

81 Sample family: positive braids σ 4 2 σ3 1 σ3 3 σ3 2 σ4 3 Theorem (F Kalfagianni Purcell) Suppose that K is the closure of a positive braid b = σ r 1 i1 σ r 2 i2 σ r k i k, where r j 3 for all j. In other words, there are k twist regions, each with at least 3 crossings. Then K is hyperbolic, and 2v 8 3 k Vol (S3 K) < 10v 3 (k 1). Similarly, v 8 (β K 1) Vol (S3 K) < 15v 3 β K 25v 3. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

82 Sample family: positive braids σ 4 2 σ3 1 σ3 3 σ3 2 σ4 3 Theorem (F Kalfagianni Purcell) Suppose that K is the closure of a positive braid b = σ r 1 i1 σ r 2 i2 σ r k i k, where r j 3 for all j. In other words, there are k twist regions, each with at least 3 crossings. Then K is hyperbolic, and 2v 8 3 k Vol (S3 K) < 10v 3 (k 1). Similarly, v 8 (β K 1) Vol (S3 K) < 15v 3 β K 25v 3. Here, v 3 = is the volume of a regular ideal tetrahedron and v 8 = is the volume of a regular ideal octahedron. The gap between the upper and lower bounds is a factor of David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

83 Sample family: Montesinos links A Montesinos knot or link is constructed by connecting n rational tangles in a cyclic fashion. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

84 Sample family: Montesinos links A Montesinos knot or link is constructed by connecting n rational tangles in a cyclic fashion. Every Montesinos link is either A or B adequate. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

85 Sample family: Montesinos links A Montesinos knot or link is constructed by connecting n rational tangles in a cyclic fashion. Every Montesinos link is either A or B adequate. Theorem (F Kalfagianni Purcell + Finlinson) Let K be an A adequate Montesinos link. Then v 8 (β K 2) Vol (S3 K). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

86 Sample family: Montesinos links A Montesinos knot or link is constructed by connecting n rational tangles in a cyclic fashion. Every Montesinos link is either A or B adequate. Theorem (F Kalfagianni Purcell + Finlinson) Let K be an A adequate Montesinos link. Then v 8 (β K 2) Vol (S3 K). If K has length at least four we get two-sided volume estimates: v 8 (max{β K, β K } 2) Vol (S3 K) < 4v 8 (β K + β K 2) + 2v 8 (#K), where #K is the number of link components of K. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

87 A coarse Volume Conjecture? Results and experimental evidence prompt: Question. Does there exist function B(K) of the coefficients of the colored Jones polynomials of a knot K, such that for hyperbolic knots, B(K) is coarsely related to hyperbolic volume Vol (S 3 K)? David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

88 A coarse Volume Conjecture? Results and experimental evidence prompt: Question. Does there exist function B(K) of the coefficients of the colored Jones polynomials of a knot K, such that for hyperbolic knots, B(K) is coarsely related to hyperbolic volume Vol (S 3 K)? Are there constants C 1 1 and C 2 0 such that for all hyperbolic K? C 1 1 B(K) C 2 Vol (S 3 K) C 1 B(K) + C 2, David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

89 A coarse Volume Conjecture? Results and experimental evidence prompt: Question. Does there exist function B(K) of the coefficients of the colored Jones polynomials of a knot K, such that for hyperbolic knots, B(K) is coarsely related to hyperbolic volume Vol (S 3 K)? Are there constants C 1 1 and C 2 0 such that for all hyperbolic K? C 1 1 B(K) C 2 Vol (S 3 K) C 1 B(K) + C 2, Volume Conjecture (Kashaev, H. Murakami-J. Murakami) predicts relations between volume and coefficients of CJP David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

90 A coarse Volume Conjecture? Results and experimental evidence prompt: Question. Does there exist function B(K) of the coefficients of the colored Jones polynomials of a knot K, such that for hyperbolic knots, B(K) is coarsely related to hyperbolic volume Vol (S 3 K)? Are there constants C 1 1 and C 2 0 such that for all hyperbolic K? C 1 1 B(K) C 2 Vol (S 3 K) C 1 B(K) + C 2, Volume Conjecture (Kashaev, H. Murakami-J. Murakami) predicts relations between volume and coefficients of CJP Proven results and stabilization properties of CJP prompt more guided speculations as to where one might look for B(K). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

91 2 edge loops and I bundles of S 3 \\S A Every 2 edge loop in G A gives rise to a disk D that intersects K twice a essential product disk (EPD) in the complement of the state surface S A. S A S A David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

92 2 edge loops and I bundles of S 3 \\S A Every 2 edge loop in G A gives rise to a disk D that intersects K twice a essential product disk (EPD) in the complement of the state surface S A. S A S A To find Guts(S 3 \\S A ), start with S 3 \\S A and remove I bundle pieces. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

93 2 edge loops and I bundles of S 3 \\S A Every 2 edge loop in G A gives rise to a disk D that intersects K twice a essential product disk (EPD) in the complement of the state surface S A. S A S A To find Guts(S 3 \\S A ), start with S 3 \\S A and remove I bundle pieces. When we remove and EPD from S 3 \\S A, Euler number χ(s 3 \\S A ) goes up by 1. Removing a redundant edge from G A also increases χ(g A ) by 1. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

94 2 edge loops and I bundles of S 3 \\S A Every 2 edge loop in G A gives rise to a disk D that intersects K twice a essential product disk (EPD) in the complement of the state surface S A. S A S A To find Guts(S 3 \\S A ), start with S 3 \\S A and remove I bundle pieces. When we remove and EPD from S 3 \\S A, Euler number χ(s 3 \\S A ) goes up by 1. Removing a redundant edge from G A also increases χ(g A ) by 1. Initially, before the cutting, χ(g A ) = χ(s A ) = χ(s 3 \\S A ). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

95 2 edge loops and I bundles of S 3 \\S A Every 2 edge loop in G A gives rise to a disk D that intersects K twice a essential product disk (EPD) in the complement of the state surface S A. S A S A To find Guts(S 3 \\S A ), start with S 3 \\S A and remove I bundle pieces. When we remove and EPD from S 3 \\S A, Euler number χ(s 3 \\S A ) goes up by 1. Removing a redundant edge from G A also increases χ(g A ) by 1. Initially, before the cutting, χ(g A ) = χ(s A ) = χ(s 3 \\S A ). We prove that the maximal I bundle of S 3 \\S A is spanned by EPD s that correspond to 2 edge loops in G A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

96 2 edge loops and I bundles of S 3 \\S A Every 2 edge loop in G A gives rise to a disk D that intersects K twice a essential product disk (EPD) in the complement of the state surface S A. S A S A To find Guts(S 3 \\S A ), start with S 3 \\S A and remove I bundle pieces. When we remove and EPD from S 3 \\S A, Euler number χ(s 3 \\S A ) goes up by 1. Removing a redundant edge from G A also increases χ(g A ) by 1. Initially, before the cutting, χ(g A ) = χ(s A ) = χ(s 3 \\S A ). We prove that the maximal I bundle of S 3 \\S A is spanned by EPD s that correspond to 2 edge loops in G A. If this correspondence is bijective, χ(guts ) = χ(s A ) + #EPDs = χ(g A extra edges) = χ(g A ). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

97 Topology of β K : most general form A 2 edge loop in G A may correspond to multiple product disks, some of which are complex. The number of complex disks is E c 0. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

98 Topology of β K : most general form A 2 edge loop in G A may correspond to multiple product disks, some of which are complex. The number of complex disks is E c 0. Theorem (F Kalfagianni Purcell) Let D(K) be an A adequate diagram. Then the state surface S A satisfies χ(guts (S 3 K, S A )) E c = χ(g A ) = 1 β K, David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

99 Topology of β K : most general form A 2 edge loop in G A may correspond to multiple product disks, some of which are complex. The number of complex disks is E c 0. Theorem (F Kalfagianni Purcell) Let D(K) be an A adequate diagram. Then the state surface S A satisfies χ(guts (S 3 K, S A )) E c = χ(g A ) = 1 β K, Under favorable conditions (positive braids, long Montesinos links, 3-braids), we get a diagram for which E c = 0, hence χ(guts ) = 1 β. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

100 Topology of β K : most general form A 2 edge loop in G A may correspond to multiple product disks, some of which are complex. The number of complex disks is E c 0. Theorem (F Kalfagianni Purcell) Let D(K) be an A adequate diagram. Then the state surface S A satisfies χ(guts (S 3 K, S A )) E c = χ(g A ) = 1 β K, Under favorable conditions (positive braids, long Montesinos links, 3-braids), we get a diagram for which E c = 0, hence χ(guts ) = 1 β. Open problem: for each A adequate link, is there a diagram with E c = 0? David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

101 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

102 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. For alternating links, this is Menasco s polyhedral decomposition: David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

103 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. For alternating links, this is Menasco s polyhedral decomposition: The two polyhedra are balloons above and below projection plane. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

104 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. For alternating links, this is Menasco s polyhedral decomposition: The two polyhedra are balloons above and below projection plane. Faces are regions of the diagram. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

105 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. For alternating links, this is Menasco s polyhedral decomposition: The two polyhedra are balloons above and below projection plane. Faces are regions of the diagram. Edges are at crossings, 4 valent. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

106 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. For alternating links, this is Menasco s polyhedral decomposition: The two polyhedra are balloons above and below projection plane. Faces are regions of the diagram. Edges are at crossings, 4 valent. Vertices are ideal (at infinity, on K ). David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

107 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. For alternating links, this is Menasco s polyhedral decomposition: The two polyhedra are balloons above and below projection plane. Faces are regions of the diagram. Edges are at crossings, 4 valent. Vertices are ideal (at infinity, on K ). Faces are checkerboard colored. The union of all the shaded faces is a checkerboard surface S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

108 Tool for the proof: a nice polyhedral decomposition Our results are proved using normal surface theory in a suitable polyhedral decomposition of the surface complement S 3 \\S A. For alternating links, this is Menasco s polyhedral decomposition: The two polyhedra are balloons above and below projection plane. Faces are regions of the diagram. Edges are at crossings, 4 valent. Vertices are ideal (at infinity, on K ). Faces are checkerboard colored. The union of all the shaded faces is a checkerboard surface S A. Hence, gluing along white faces only produces a decomposition of S 3 \\S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

109 Polyhedral decomposition of the surface complement Our surface S A is layered below the plane of projection. We need more balloons to subdivide S 3 \\S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

110 Polyhedral decomposition of S 3 \\S A : 3 cells 3 cells: One upper 3 cell, above the plane of projection. One lower 3 cell for each non-trivial component of complement of state circles in A resolution. (Innermost disks are trivial.) Two nontrivial components David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

111 Polyhedral decomposition of S 3 \\S A : 3 cells 3 cells: One upper 3 cell, above the plane of projection. One lower 3 cell for each non-trivial component of complement of state circles in A resolution. (Innermost disks are trivial.) Two nontrivial components David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

112 Polyhedral decomposition of S 3 \\S A : 3 cells 3 cells: One upper 3 cell, above the plane of projection. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

113 Polyhedral decomposition of S 3 \\S A : 3 cells 3 cells: One upper 3 cell, above the plane of projection. One lower 3 cell for each non-trivial component of complement of state circles in A resolution. (Innermost disks are trivial.) Two nontrivial components David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

114 Polyhedral decomposition of S 3 \\S A : faces Faces are checkerboard colored, and come in two distinct flavors: Portions of a 3 cell meeting S A. These faces are shaded. Disks lying slightly below the plane of projection, with boundary on S A. One disk for each region of the graph H A (state circles and red edges). These faces are white. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

115 Polyhedral decomposition of S 3 \\S A : faces Faces are checkerboard colored, and come in two distinct flavors: Portions of a 3 cell meeting S A. These faces are shaded. Disks lying slightly below the plane of projection, with boundary on S A. One disk for each region of the graph H A (state circles and red edges). These faces are white. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

116 Polyhedral decomposition of S 3 \\S A : faces David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37

117 Polyhedral decomposition of S 3 \\S A : faces Faces are checkerboard colored, and come in two distinct flavors: Portions of a 3 cell meeting S A. These faces are shaded. Disks lying slightly below the plane of projection, with boundary on S A. One disk for each region of the graph H A (state circles and red edges). These faces are white. All polyhedra are glued to the upper polyhedron, along white faces only. David Futer, Effie Kalfagianni, and Jessica S. Purcell () June / 37