The geometry of state surfaces and the colored Jones polynomials

Transcription

1 The geometry of state surfaces and the colored Jones polynomials joint with D. Futer and J. Purcell December 8, 2012 David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

2 Given: Diagram of a knot or link 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 () December 8, / 15

3 Example states Link diagram All A state All B state Above: H A and H B. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

4 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 (Dasbach-Futer-K-Lin-Stoltzfus, 2006). David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

5 Colored Jones polynomial prelims For a knot K we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β n tkn+1 + α n tkn. 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 () December 8, / 15

6 Colored Jones polynomial prelims For a knot K we write its n-colored Jones polynomial: J K,n (t) := α n t mn + β n t mn β n tkn+1 + α n tkn. 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! David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

7 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. A or B-adequate: all alternating knots, Montesinos knots, positive knots, arborescent knots, blackboard cables of adequate knots, all but a handful of prime knots up to 12 crossings... David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

8 Colored Jones polynomials for A-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. (Lickorish-Thistlethwaite) Coefficients α n = ±1 are independent of n: α K := α n. (L-T) k n :=min deg J K,n (t) quadratic polynomial in n; can be calculated explicitly. (Dasbach-Lin/ Stoimenow) Coefficients β n are independent of n. We have β K := β n = 1 χ(g A ). David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

9 Colored Jones polynomials for A-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. (Lickorish-Thistlethwaite) Coefficients α n = ±1 are independent of n: α K := α n. (L-T) k n :=min deg J K,n (t) quadratic polynomial in n; can be calculated explicitly. (Dasbach-Lin/ Stoimenow) Coefficients β n are independent of n. We have β K := β n = 1 χ(g A ). Similar properties for B-adequate. Note: In example above G A is a tree so β K = 0. The surface S A is fiber of the knot complement (later). David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

10 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 () December 8, / 15

11 When are state surfaces essential? Essential=incompressible, -incompressible (π 1 -injective). ( Warning.If S A is non-orientabe use double: N(S A ) N( S A ).) The only trouble. If H A has edge with both endpoints on a single state circle, then we form boundary compression disk: Theorem (Ozawa, Futer-K-Purcell)The following are equivalent: H A has no edge with both endpoints on a single state circle S A is essential. Ozawa proof is different; uses Murasugi sum arguments. We see more about S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

12 Boundary slopes and CJP 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, k n :=min deg J K,n (t). 4 bdry slope of S A = lim n n 2 k n, Similar statement for B-adequate. Slopes Conjecture. (Garoufalidis) For every knot K the sequence { 4 k n 2 n } n has finitely many cluster points, each of which is a boundary slope of K. (Garoufalidis-Dunfield) Verified conjecture for a class of knots that are not A-adequate. ( Degree of CJP was done by computer calculation). David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

13 CJP detects fibers in state surfaces Recall the reduced state graph G A corresponding to an A-adequate diagram and β K the stabilized penultimate coefficient of the CJP of the corresponding knot K. Theorem (FKP) The following are equivalent: The complement S 3 K fibers over S 1 with fiber S A. The reduced graph G A is a tree. We have β K = 0. Remark If β K 0 then S A doesn t lift to a fiber in any finite cover!. Next. If K is hyperbolic, β K determines the geometric type of S A in the Thurston-Bonahon trichotomy. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

14 Essential surfaces in cusped hyperbolic 3-manifolds M be a compact, hyperbolic 3 manifold with boundary consisting of tori, and let S be a properly embedded essential surface in M. We have a faithful representation ρ : π 1 (S) π 1 (M) PSL(2, C). accidental parabolic on S: a free homotopy class of a closed curve not parallel on S but can be homotoped to the boundary of M. An accidental parabolic is a non-peripheral element of π 1 (S) that is is mapped by ρ to a parabolic in π 1 (M). A surface S with accidental parabolics is called accidental. S is called quasifuchsian if the embedding S M lifts to a topological plane in H 3 whose limit set Λ H 3 is a topological circle. (We permit the two disks bounded by the Jordan curve Λ will be be interchanged by isometries corresponding to π 1 (S).) The surface S is a semi-fiber if it is a fiber in M or covered by a fiber in a two-fold cover of M. If S is a semi-fiber but not a fiber, we call it a strict semi-fiber. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

15 Types of surfaces: A trichotomy Every properly embedded, essential surface S in a hyperbolic 3 manifold M falls into exactly one of the three types: S is either a semi-fiber, or accidental, or quasifuchsian. (Thurston, Bonahon). Historical Remarks.Types of surfaces in hyperbolic link complements. 1 No alternating link complement contains embedded quasifuchsian closed surfaces. (Menasco- Reid). 2 There are closed, immersed quasifuchsian surfaces in any hyperbolic link complement (Masters-Zhang). 3 There are closed accidental surfaces in a wide range of link complements ( Finkelstein- Moriah, Wu...) 4 Minimum genus Seifert surfaces are not accidental ( Cooper-Long, Fenley). 5 There are hyperbolic knots with accidental Seifert surfaces (Tsutsumi...). 6 Checkerboard surfaces in alternating link complements are not virtual fibers (Wise); they are always quasifuchsian (Adams, Futer-K.-Purcell). David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

16 CJP determines the geometric type of state surfaces Recall the reduced state graph G A corresponding to an A-adequate diagram and β K the stabilized penultimate coefficient of the CJP of the corresponding knot K. Theorem (FKP) Suppose K is hyperbolic. Then, β K determines the geometric type of the all A surface S A, as follows: S A is a fiber in S 3 K iff β K = 0 or equivalently G A is a tree. S A is quasifuchsian iff β K 0 or equivalently G A is not a tree. Remark. We have large families of quasifuchsian that fit nicely with recent work of Thistlethwaite-Tsvietkova, who proposed an algorithm to construct the hyperbolic structure on a link complement directly from a diagram. Their algorithm works whenever a link diagram admits a non-accidental state surface, which is exactly what our results ensure (see next talk!!). David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

17 A nice polyhedral decomposition. 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 (what remains from (S 3 K) after cutting along S A ). Result: Starting with an A-adequate diagram, we obtain an ideal polyhedral decomposition of S 3 \\S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

18 A nice polyhedral decomposition. 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 (what remains from (S 3 K) after cutting along S A ). Result: Starting with an A-adequate diagram, we obtain an ideal polyhedral decomposition of S 3 \\S A. Properties of resulting decomposition: Polyhedra: 3-balls with a 4-valent graph on their boundary: the regions in the complement of the graph are the faces. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

19 A nice polyhedral decomposition. 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 (what remains from (S 3 K) after cutting along S A ). Result: Starting with an A-adequate diagram, we obtain an ideal polyhedral decomposition of S 3 \\S A. Properties of resulting decomposition: Polyhedra: 3-balls with a 4-valent graph on their boundary: the regions in the complement of the graph are the faces. All faces are simply connected. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

20 A nice polyhedral decomposition. 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 (what remains from (S 3 K) after cutting along S A ). Result: Starting with an A-adequate diagram, we obtain an ideal polyhedral decomposition of S 3 \\S A. Properties of resulting decomposition: Polyhedra: 3-balls with a 4-valent graph on their boundary: the regions in the complement of the graph are the faces. All faces are simply connected. Checkerboard colored (shaded faces and white faces). David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

21 A nice polyhedral decomposition. 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 (what remains from (S 3 K) after cutting along S A ). Result: Starting with an A-adequate diagram, we obtain an ideal polyhedral decomposition of S 3 \\S A. Properties of resulting decomposition: Polyhedra: 3-balls with a 4-valent graph on their boundary: the regions in the complement of the graph are the faces. All faces are simply connected. Checkerboard colored (shaded faces and white faces). Ideal vertices are 4 valent; they lie on parabolic locus. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

22 A nice polyhedral decomposition. 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 (what remains from (S 3 K) after cutting along S A ). Result: Starting with an A-adequate diagram, we obtain an ideal polyhedral decomposition of S 3 \\S A. Properties of resulting decomposition: Polyhedra: 3-balls with a 4-valent graph on their boundary: the regions in the complement of the graph are the faces. All faces are simply connected. Checkerboard colored (shaded faces and white faces). Ideal vertices are 4 valent; they lie on parabolic locus. Combinatorics determined by graph H A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

23 A nice polyhedral decomposition. 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 (what remains from (S 3 K) after cutting along S A ). Result: Starting with an A-adequate diagram, we obtain an ideal polyhedral decomposition of S 3 \\S A. Properties of resulting decomposition: Polyhedra: 3-balls with a 4-valent graph on their boundary: the regions in the complement of the graph are the faces. All faces are simply connected. Checkerboard colored (shaded faces and white faces). Ideal vertices are 4 valent; they lie on parabolic locus. Combinatorics determined by graph H A. Use these polyhedra and normal surface theory to study geometry of S A. David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15

24 A word on the proofs By the Thurston-Bonahon trichotomy, S A is never a semi-fiber (FKP, 2010). S A doesn t contain any accidental parabolics (FKP, 2012). A word on ruling out parabolics: Existence of an accidental parabolic implies the existence of an essential, embedded annulus A in M A = S 3 \\S A, with one -component on S A and one on the parabolic locus. Put the embedded annulus in normal form and analyze its intersections with the polyhedral decomposition of M A. - Get contradiction annulus cannot exist if link to start with K, is hyperbolic. Example. Suppose K is a knot (one component): On any white face W, of the decomposition, Σ intersect in arcs such that each arc in A W runs from an ideal vertex of W to a side of W ; and A W passes through every ideal vertex of W. Then an arc in A W runs from an ideal vertex to an adjacent side of W. Contradiction to normal form! David Futer, Effie Kalfagianni, and Jessica S. Purcell () December 8, / 15