AddressDepartment of Mathematics, Room 1S209College of Staten Island (CUNY) 2800 Victory Boulevard Staten Island, NY 10314 (718) 9823615

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 ribbongraph polynomial invariants.
We identify all hyperbolic knots whose complements are in the census of orientable onecusped 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 nbraid L with a full positive twist and with k negative crossings, 0\leq k \leq n, we determine the first nk+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 nk between the first two nonzero coefficients of (1t^2)V_L(t). For a closed positive nbraid with a full positive twist, we extend our results to the colored Jones polynomials. For N>n1, we determine the first n(N1)+1 terms of the normalized Nth 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. Tlinks are a natural generalization,
given by repeated positive twisting. We establish a onetoone
correspondence between positive braid representatives of Lorenz links
and Tlinks, so Lorenz links and Tlinks 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 Tlinks, and establishes many new
results for Tlinks, 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_2term of this spectral sequence is a matroid invariant and hence invariant under mutation.
Quasialternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasialternating link L gives rise to an infinite family of quasialternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are quasialternating, 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 BollobasRiordanTutte polynomial is a threevariable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasitree 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 quasitree expansion of the BollobasRiordanTutte polynomial.
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasitree 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 quasitrees 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 twistbracket 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 CWcomplex whose rational cohomology algebra contains Vassiliev invariants of knots in a 3manifold. 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.