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Journals and Conferences
The colored Jones polynomial is a function JK : N −→ Z[t, t] associated with a knot K in 3-space. We will show that for an alternating knot K the absolute values of the first and the last three leading coefficients of JK(n) are independent of n when n is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading… (More)
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobás–Riordan– Tutte polynomial generalizes the Tutte polynomial of graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we… (More)
The Volume conjecture claims that the hyperbolic Volume of a knot is determined by the colored Jones polynomial. The purpose of this article is to show a Volume-ish theorem for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the Volume and certain sums of coefficients of the Jones polynomial is… (More)
Every link in R can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.
A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link. Furthermore,… (More)
There were many attempts to settle the question whether there exist non-trivial knots with trivial Jones polynomial. In this paper we show that such a knot must have crossing number at least 18. Furthermore we give the number of prime alternating knots and an upper bound for the number of prime knots up to 17 crossings. We also compute the number of… (More)
We prove that the dimension of the space of primitive Vassiliev invariants of degree n grows as n tends to infinity faster than ec √
We show that a certain linear representation of the singular braid monoid SB3 is faithful. Furthermore we will give a second group theoretically motivated solution to the word problem in SB3.
Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate infinite families of links with the same… (More)
We will explore the experimental observation that on the set of knots with bounded crossing number, algebraically independent Vassiliev invariants become correlated, as noticed first by S. Willerton. We will see this through the value distribution of the Jones polynomial at roots of unit. As the degree of the roots of unit is getting larger, the higher… (More)