Address
Department of Mathematics, Room 1S-209College of Staten Island (CUNY)
2800 Victory Boulevard
Staten Island, NY 10314
Phone: (718) 982-3615
email: ikofman
math.csi.cuny.edu
AddressDepartment of Mathematics, Room 1S-209College of Staten Island (CUNY) 2800 Victory Boulevard Staten Island, NY 10314 Phone: (718) 982-3615
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MTH 229 final exam will be in 1S-103 on Monday, December 22, 2:30--4:25pm. Sample Math 229 FINAL EXAM, sample midterm exam.
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.
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)
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.