Determinant density and the Vol-Det Conjecture

Determinant density and the Vol-Det Conjecture Ilya Kofman College of Staten Island and The Graduate Center City University of New York (CUNY) Joint work with Abhijit Champanerkar and Jessica Purcell December 16, 2016 Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 1/39

Open problem in geometric topology How to relate the two perspectives on, say, the knot 8 18... Diagrammatic invariants Determinant Jones polynomial etc. Geometric invariants Hyperbolic volume A polynomial etc. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 2/39

Main idea of this talk: Easier to do this for biperiodic alternating links We consider biperiodic alternating links from three points of view: 1 Geometric (volume density) 2 Diagrammatic (determinant density) 3 Asymptotic, as limits of sequences of finite hyperbolic links Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 3/39

Volume density Volume density of K is defined as Vol(K)/c(K). Theorem (Adams) 0 < Vol(K)/c(K) < v oct. Can decompose S 3 K into octahedra, one octahedron at each crossing: The volume density spectrum Spec vol is the derived set of the set of all volume densities of links. Volume bounds =) Spec vol [0, v oct ] Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 4/39

Determinant density The knot determinant was one of the first computable knot invariants (computable = not of the form minimize something over all diagrams ) det(k) = det(m + M T ), M =Seifertmatrix = H 1 ( 2 (K); Z), 2 = 2 fold branched cover of K = V K ( 1) = K ( 1), V K, K = Jones, Alexander poly = #spanning trees (G K ), G K = Tait graph of alternating K Determinant density of K is defined as 2 log det(k)/c(k). Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 5/39

Determinant density Conjecture 1 (Kenyon 96) For any finite planar graph G, log (G) e(g) apple 2C, where C 0.916 is Catalan s constant, e(g) = # edges of G, and (G) =# spanning trees of G. If G = G K then e(g) =c(k), det(k) apple (G K ), and 4C = v oct. Conjecture 2 For any knot or link K, thedeterminantdensitysatisfies 2 log det(k) c(k) apple v oct. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 6/39

Determinant density The determinant density spectrum Spec det is the derived set of the set of all determinant densities. Conjecture 2 =) Spec det [0, v oct ] Theorem (Stoimenow 07) Let 1.8393 be the real positive root of x 3 x 2 x 1 = 0. Then for any knot or link K 2 log det(k) c(k) apple 2 log 3.8288. Note: v oct 3.66 Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 7/39

Determinant and hyperbolic volume Dunfield (2000) suggested a relationship between det(k) andvol(s 3 K): Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 8/39

Determinant and hyperbolic volume The Volume Conjecture provided another context: 2 log hki N lim = Vol(K) N!1 N hki N := J N(K;exp(2 i/n)),sothen =2termis2 log det(k) /2. J N ( ;exp(2 i/n)) Stoimenow ( 07) proved a coarse relationship for any hyperbolic alternating link K: There are constants C 1, C 2 > 0, such that 2 1.0355 Vol(K) apple det(k) apple C1 c(k) C2 Vol(K) Vol(K) Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 9/39

Vol-Det Conjecture Conjecture (Vol-Det Conjecture): For any alternating hyperbolic link K, Vol(K) < 2 log det(k). Verified for all alternating knots apple 16 crossings. (Champanerkar-K-Purcell) 2 is sharp. Conjecture Vol(K) < 2 log rank( e H, (K)) for any hyperbolic knot K. Verified for all non-alternating knots apple 15 crossings. Remark Let K be a reduced alternating link diagram, and let K 0 be obtained by changing any proper subset of crossings of K. (Champanerkar-K-Purcell) det(k 0 ) < det(k). Conjecture Vol(K 0 ) < Vol(K). Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 10 / 39

Vol-Det Conjecture Conjecture (Vol-Det Conjecture): For any alternating hyperbolic link K, Vol(K) < 2 log det(k). Verified for all alternating knots apple 16 crossings. (Champanerkar-K-Purcell) 2 is sharp. i.e., if <2 then there exist alternating hyperbolic knots K such that log det(k) < Vol(K) Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 10 / 39

Infinite square weave The infinite square weave W is the infinite alternating link whose diagram projects to the square lattice. The Z 2 quotient of R 3 W is a link complement in a thickened torus. T 2 I = S 3 Hopf link, so it s also the complement of link ` in S 3 with Hopf sublink. ` has complete hyperbolic structure with four regular ideal octahedra. S 3 Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 11 / 39

Følner convergence Finite connected subgraphs G n G form a Følner sequence for G if G n G n+1 (nested) [G n = G (exhaustive) @G n lim = 0 n!1 G n (Følner) A sequence of alternating links {K n } Følner converges almost everywhere to the biperiodic alternating link L if the respective projection graphs {G(K n )} and G(L) satisfy the following two conditions: (i) 9G n G(K n ) that form a toroidal Følner sequence for G(L) i.e. G n is Følner sequence for G(L), and G n G(L) \ n (ii) lim G n /c(k n ) = 1. n!1 We denote this by K n F!L. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 12 / 39

Geometrically and diagrammaticaly maximal knots Theorem 1 (Champanerkar-K-Purcell) Let K n be any hyperbolic alternating links with no cycles of 2 tangles. F Vol(K n ) K n!w =) lim = v oct = lim n!1 c(k n ) n!1 2 log det(k n ). c(k n ) OK not OK Corollary v oct 2 Spec vol \ Spec det. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 13 / 39

Density Spectra Theorem (Adams-Calderon-Jiang-Kastner-Kehne-Mayer-Smith 15) For any x 2 [0, v oct ], there exist sequences of hyperbolic knots K n such that Vol(K n ) lim = x = lim n!1 c(k n ) n!1 Corollary (Adams et.al. 15, Burton 15) (a) [0, v oct ]=Spec vol (b) [0, v oct ] Spec det 2 log det(k n ). c(k n ) Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 14 / 39

Theorem 1 (Champanerkar-K-Purcell) Let K n be any hyperbolic alternating links with no cycles of 2 tangles. F Vol(K n ) K n!w =) lim = v oct = lim n!1 c(k n ) n!1 Volume density of W = Vol(T 2 I L)/c(L) =v oct. 2 log det(k n ). c(k n ) Conjecture (Volume density convergence) Let L be any biperiodic alternating link. Let L = L/ be the toroidally alternating quotient link. F Vol(K n ) K n!l =) lim n!1 c(k n ) = Vol(T 2 I L). c(l) Question What analogous toroidal invariant is the limit for the determinant density? Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 15 / 39

Spanning tree entropy Square weave W Tait graph G W = square grid (Burton-Pemantle, Shrock-Wu) Spanning tree entropy of W: G n = n n square grid, #spanning trees (G n ), Catalan s C 0.916 log (G n ) lim n!1 n 2 =4C = v oct (Kasteleyn) Growth rate of the number of dimers Z(G n ): log Z(G n ) lim n!1 n 2 =4C = v oct Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 16 / 39

Dimers A dimer covering of a graph G is a set of edges that covers every vertex exactly once, i.e. a perfect matching. The dimer model is the study of the set of dimer coverings of G. Let Z(G) = # dimer coverings of G. Theorem (Kasteleyn 63) If G is a balanced bipartite planar graph, where K is a Kasteleyn matrix. Z(G) =det(k), Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 17 / 39

Dimers and Spanning trees For any finite plane graph G, overlay G and its dual G, delete a vertex of G and G (in the unbounded face) and delete all incident edges to get balanced bipartite graph G b. G G [ G G b A dimer on G b Theorem (Burton-Pemantle 93, Propp 02) (G) =Z(G b ). Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 18 / 39

Biperiodic bipartite graphs Biperiodic link L!Biperiodic bipartite graph G L [ G L. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 19 / 39

Kasteleyn matrix for toroidal dimer model Let G b be a finite balanced bipartite toroidal graph. Kasteleyn matrix K(z, w) for toroidal dimer model on G b is defined by: 1 Choose Kasteleyn weighting (signs on edges, such that each face with 0 mod 4 edges has odd # of signs). 2 Choose oriented scc s z, w on T 2 that are basis of H 1 (T 2 ). Orient each edge e from black to white. Let µ e = z z e w w e. 3 Order the black and white vertices. Then K(z, w) isthe B W adjacency matrix with entries ±µ e. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 20 / 39

Kasteleyn matrix for toroidal dimer model K(z, w) = apple 1 1/z 1+w 1+1/w 1+z Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 21 / 39

Mahler measure Mahler measure of polynomial p(z) isdefinedas m(p(z)) = 1 Z log p(z) dz X Jensen = 2 i S 1 z i roots of p max{log i, 0}. 2-variable Mahler measure: m(p(z, w)) = 1 (2 i) 2 Smyth s remarkable formula: Z log p(z, w) dz T 2 z dw w. 2 m(1 + x + y) = 3p 3 2 L( 3, 2) = Vol( ) Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 22 / 39

Toroidal dimer model Let G b be a biperiodic balanced bipartite planar graph, which is invariant under translations by 2-dim lattice. The characteristic polynomial of the toroidal dimer model on G b is p(z, w) =detk(z, w). Theorem (Kenyon-Okounkov-She eld 06) Let G b be biperiodic balanced bipartite graph, and G n = G b /n be a toroidal exhaustion of G b.then the partition function satisfies: log Z(G b 1 ):= lim n!1 n 2 log Z(G n)=m(p(z, w)). Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 23 / 39

Biperiodic alternating links Theorem (Champanerkar-K) Let L be any biperiodic alternating link with Tait graph G L.LetL = L/ be the toroidally alternating quotient link. Let p(z, w) be the characteristic polynomial for the toroidal dimer model on G b = G L [ G L. F 2 log det(k n ) K n!l =) lim = n!1 c(k n ) 2 m(p(z, w)). c(l) The RHS is called the determinant density of L. Idea of proof: The following limits are equal: 1 Spanning tree model on G L, i.e. limit of spanning tree entropies of planar exhaustions of G L. 2 Toroidal dimer model on biperiodic, bipartite graph G b = G L [ G L, i.e. limit of dimer entropies of the toroidal exhaustions of G b. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 24 / 39

Semi-regular Euclidean tilings Let G be a 4-valent, semi-regular biperiodic tiling of the Euclidean plane. Let L be the alternating link such that G = G(L). Examples: Square weave W and square lattice G(W): Triaxial link L and trihexagonal lattice G(L): Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 25 / 39

More examples of 4-valent semi-regular Euclidean tilings https://en.wikipedia.org/wiki/euclidean tilings by convex regular polygons Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 26 / 39

Geometry of semi-regular Euclidean tilings Theorem (Champanerkar-K-Purcell) Let L be any biperiodic alternating link whose projection graph G(L) is a semi-regular Euclidean tiling. Then L has a complete hyperbolic structure with volume density: 10av tet + bv oct per fundamental domain, where a = #hexagons, and b = #squares in the fundamental domain. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 27 / 39

Geometry of semi-regular Euclidean tilings Example: Rhombitrihexagonal link L, and quotient link L in T 2 I Vol(T 2 I L) = 10v tet +3v oct. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 28 / 39

Example: Infinite square weave W W & L G b W & G b L Kasteleyn weighting Vol(T 2 I L) =2v oct =7.32772... Characteristic polynomial: p(z, w) = (4 + 1/w + w +1/z + z). 2 log Z =2 m(p(z, w)) = 2 v oct exactly (Boyd & Rodriguez-Villegas) 2 log det(k n ) 2 m(p(z, w)) lim = = Vol(T 2 I L) Vol(K n ) = lim n!1 c(k n ) 2 c(l) n!1 c(k n ). Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 29 / 39

Example: Triaxial link L G(L) &L G b L & G b L Kasteleyn weighting Vol(T 2 I L) = 10 v tet = 10.14941... p triaxial (z, w) =6 1/w w 1/z z w/z z/w. 2 log Z =2 m(p(z, w)) = 10 v tet exactly (Boyd & Rodriguez-Villegas) 2 log det(k n ) 2 m(p(z, w)) lim = = Vol(T 2 I L) Vol(K n ) = lim n!1 c(k n ) 3 c(l) n!1 c(k n ). Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 30 / 39

Example: Rhombitrihexagonal link L G(L) L GL b Vol(T 2 I L) = 10 v tet +3v oct = 21.14100... p(z, w) =6(6 1/w w 1/z z w/z z/w) =6p triaxial (z, w) 2 m(p(z, w)) = 2 log 6 + 10v tet = 21.40737... Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 31 / 39

Vol-Det Conjecture for an infinite family of knots Conjecture (Vol-Det Conjecture): For any alternating hyperbolic link K, Vol(K) < 2 log det(k). Idea: Use biperiodic alternating links to obtain infinite families of links satisfying the Vol-Det Conjecture. This is possible if for K n F!L, e.g. Rhombitrihexagonal link L. Vol(K n ) 2 log det(k n ) lim < lim n!1 c(k n ) n!1 c(k n ) Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 32 / 39

Vol-Det Conjecture for an infinite family of knots Theorem (Champanerkar-K-Purcell 15) Let L be the Rhombitrihexagonal link and let K n be any sequence of F alternating links such that K n!l.thenkn satisfies the Vol-Det Conjecture for almost all n. Proof: Let L be the Rhombitrihexagonal link, and let K n F!L. 2 log det(k n ) 6 lim n!1 c(k n ) = 2 m(p(z, w)) = 10 v tet +2 log(6) = 21.40737... > 10v tet +3v oct = 21.14100... = Vol(T 2 I L) Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 33 / 39

Vol-Det Conjecture for an infinite family of knots Final step uses Følner convergence, K n F!L: The geodesic checkerboard polyhedron of K n has the rhombitrihexagonal geometry almost everywhere, hence: Vol(K n ) c(k n ) apple Vol(T 2 I L) c(l) + O @Gn (v tet ) G n The result now follows because Følner condition =) @G n G n! 0. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 34 / 39

Remark 1 2 log det(k n ) Vol(K n ) The proof above fails when lim = lim n!1 c(k n ) n!1 c(k n ). e.g., the square weave W, and the triaxial link L F We checked numerically for weaving knots K n!w with hundreds of crossings that the Vol-Det Conjecture does hold. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 35 / 39

Remark 2 Corollary (Champanerkar-K-Purcell) T Let K [ B be any augmented alternating link, and let K n denote the n periodic alternating link with quotient K, formed by taking n copies of a tangle T.Then T T T T T T Vol(K n ) lim = n!1 c(k n ) Vol(K [ B) 2 log det(k n ), and lim = c(k) n!1 c(k n ) 2 log det(k). c(k) Corollary + Vol-Det Conjecture =) If K [ B is any augmented alternating link, then Vol(K) < Vol(K [ B) apple 2 log det(k) Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 36 / 39

Remark 3: Isoradial toroidal dimer model If G is isoradial and biperiodic, Z(G b ) can be expressed geometrically. Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 37 / 39

Remark 3: Isoradial toroidal dimer model Theorem (Kenyon 02, de Tilière 07) 2 log Z(G b 1 ) = 2 lim n!1 n 2 log Z(G n)=2 m(p(z, w)) = X e2f.d. (2 ( e ) + 2 e log(2 sin e )) = Vol f.d. (P(G b )) + X e2f.d. 2 e log(2 sin e ) where the sum is over all edges e in the fundamental domain (f.d.), e = 1 2 rhombus angle at e, andp(g b ) is an ideal hyperbolic polyhedron: Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 38 / 39

This is still work in progress... Ilya Kofman (CUNY) Determinant density and the Vol-Det Conjecture 39 / 39