Professor Department of Mathematics College of Staten Island, The City University of New York (CUNY) 2800 Victory Boulevard Staten Island, NY 10314 Office: 1S-209 Phone: 718-982-3615 ikofmanmath.csi.cuny.edu |
Doctoral Faculty Mathematics Program The Graduate Center, The City University of New York (CUNY) 365 Fifth Avenue New York, NY 10016 Office: 4214.02 Phone: 212-817-8540 |
The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in S^3. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of alternating links. We conjecture a new lower bound for the Mahler measure of certain two-variable polynomials in terms of volumes of hyperbolic regular ideal bipyramids. Associating each polynomial to a toroidal link using the toroidal dimer model, we show that every polynomial which satisfies this conjecture with a strict inequality gives rise to many infinite families of alternating links satisfying the Vol-Det Conjecture. We prove this new conjecture for six toroidal links by rigorously computing the Mahler measures of their two-variable polynomials.
A chapter on Turaev surfaces to appear in the Concise Encyclopedia of Knot Theory.
A biperiodic alternating link has an alternating quotient link in the thickened torus. In this paper, we focus on semi-regular links, a class of biperiodic alternating links whose hyperbolic structure can be immediately determined from a corresponding Euclidean tiling. Consequently, we determine the exact volumes of semi-regular links. We relate their commensurability and arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor, we show there exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field. We show that only two semi-regular links have totally geodesic checkerboard surfaces; these two links satisfy the Volume Density Conjecture. Finally, we give conditions implying that many additional biperiodic alternating links are hyperbolic and admit a positively oriented, unimodular geometric triangulation. We also provide sharp upper and lower volume bounds for these links.
Let L be any infinite biperiodic alternating link. We show that for any sequence of finite links that Folner converges almost everywhere to L, their determinant densities converge to the Mahler measure of the 2-variable characteristic polynomial of the toroidal dimer model on an associated biperiodic graph.
Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for weaving knots, and we prove that the infinite weave is their geometric limit.
The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios.
We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also propose related conjectures motivated by geometrically and diagrammatically maximal sequences of knots.
The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants.
We identify all hyperbolic knots whose complements are in the census of orientable one-cusped hyperbolic manifolds with eight ideal tetrahedra. We also compute their Jones polynomials.
Twisted torus knots and links are given by twisting adjacent strands of a
torus link. They are geometrically simple and contain many examples of
the smallest volume hyperbolic knots. Many are also Lorenz links.
We study the geometry of twisted torus links and related generalizations.
We determine upper bounds on their hyperbolic volumes that depend only on
the number of strands being twisted. We exhibit a family of twisted torus
knots for which this upper bound is sharp, and another family with volumes
approaching infinity. Consequently, we show there exist twisted torus
knots with arbitrarily large braid index and yet bounded volume.
For a closed n-braid L with a full positive twist and with k negative crossings, 0\leq k \leq n, we determine the first n-k+1 terms of the Jones polynomial V_L(t). We show that V_L(t) satisfies a braid index constraint, which is a gap of length at least n-k between the first two nonzero coefficients of (1-t^2)V_L(t). For a closed positive n-braid with a full positive twist, we extend our results to the colored Jones polynomials. For N>n-1, we determine the first n(N-1)+1 terms of the normalized N-th colored Jones polynomial.
A Lorenz knot is the isotopy class of any periodic orbit in the flow
on R^3 given by the Lorenz differential equations. Twisted torus
links are given by twisting a subset of strands on a closed braid
representative of a torus link. T--links are a natural generalization,
given by repeated positive twisting. We establish a one-to-one
correspondence between positive braid representatives of Lorenz links
and T--links, so Lorenz links and T--links coincide. Using this
correspondence, we identify over half of the simplest hyperbolic knots
as Lorenz knots. We show that both hyperbolic volume and the Mahler
measure of Jones polynomials are bounded for infinite collections of
hyperbolic Lorenz links. The correspondence provides unexpected
symmetries for both Lorenz links and T-links, and establishes many new
results for T-links, including new braid index formulas.
Two typos in Table 1 should be corrected: k6_35 = <6^6, 8^5> and k7_61 = <3^2, 7^10>.
It is conjectured that the Khovanov homology of a knot is invariant under mutation. In this paper, we review the spanning tree complex for Khovanov homology, and reformulate this conjecture using a matroid obtained from the Tait graph (checkerboard graph) G of a knot diagram K. The spanning trees of G provide a filtration and a spectral sequence that converges to the reduced Khovanov homology of K. We show that the E_2-term of this spectral sequence is a matroid invariant and hence invariant under mutation.
Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link L gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are quasi-alternating, and we determine the thickness of Khovanov homology for ``most'' pretzel links with arbitrarily many strands.
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a spectral sequence that converges to its homology. For alternating links, all differentials on the spanning tree complex are zero and the reduced Khovanov homology is determined by the Jones polynomial and signature. We prove some analogous theorems for (unreduced) Khovanov homology.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobas-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We generalize the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollobas-Riordan-Tutte polynomial.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram L, there is an associated ribbon graph whose quasi-trees correspond bijectively to spanning trees of the graph obtained by checkerboard coloring L. This correspondence preserves the bigrading used for the spanning tree model of Khovanov homology, whose Euler characteristic is the Jones polynomial of L. Thus, Khovanov homology can be expressed in terms of ribbon graphs, with generators given by ordered chord diagrams.
We show that if {L_n} is any infinite sequence of links with twist number \tau(L_n) and with cyclotomic Jones polynomials of increasing span, then \lim\sup \tau(L_n)=\infty. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main tool is the multivariable twist-bracket polynomial, which generalizes the Kauffman bracket to link diagrams with open twist sites.
We show that the Mahler measure of the Jones polynomial and of the colored Jones polynomials converges under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a_1,...,a_n) we show that the Mahler measure of the Jones polynomial converges if all a_i approach infinity, and approaches infinity for constant a_i if n approaches infinity, just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.
We complete the project begun by Callahan, Dean, and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these ``simple'' hyperbolic knots have high crossing number. You can now see many of them! (thanks to Rob Scharein). The published version had the last term missing from some Dowker codes in Table 4, which has now been corrected in the version on the ArXiv (with thanks to David Boyd for pointing out the error).
We construct a cubical CW-complex whose rational cohomology algebra contains Vassiliev invariants of knots in a 3-manifold. We compute the first two homotopy groups, and give conditions for Vassiliev invariants to be nontrivial in cohomology. For R^3 we show that any Vassiliev invariant coming from the Conway polynomial is nontrivial in cohomology. The cup product provides a new graded commutative algebra of Vassiliev invariants evaluated on ordered singular knots. We show how the cup product arises naturally from a cocommutative differential graded Hopf algebra of ordered chord diagrams.
We find the first approximations by Vassiliev invariants for the coefficients of the Jones polynomial and all specializations of the HOMFLY and Kauffman polynomials. Consequently, we obtain approximations of invariants arising from the homology of branched covers of links.