Related papers: A generating polynomial for the two-bridge knot wi…
For a Catalan state $C$ of a lattice crossing $L\left( m,n\right) $ with no returns on one side, we find its coefficient $C\left( A\right) $ in the Relative Kauffman Bracket Skein Module expansion of $L\left( m,n\right) $. We show, in…
Recent progress in string theory has led to a reformulation of quantum-group polynomial invariants for knots and links into new polynomial invariants whose coefficients can be understood in topological terms. We describe in detail how to…
In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to…
We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism…
Kalfagianni and Lee found two-sided bounds for the crosscap number of an alternating link in terms of certain coefficients of the Jones polynomial. We show here that we can find similar two-sided bounds for the crosscap number of Conway…
We continue the study of quantum A-polynomials -- equations for knot polynomials with respect to their coloring (representation-dependence) -- as the relations between different links, obtained by hanging additional ``simple'' components on…
In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as $f(z)f(-z)$. In this paper, we normalize the factor $f(z)$ to define the half-Conway polynomial. First, we prove that the…
This paper is about the clock number of a knot. First we define the clock number by using states of a knot defined by Kauffman. Next we show that if K is a prime knot, its clock number is greater than or equal to its crossing number.…
We develop a dimer model for the Alexander polynomial of a knot. This recovers Kauffman's state sum model for the Alexander polynomial using the language of dimers. By providing some additional structure we are able to extend this model to…
Link/knot invariants are series with integer coefficients, and it is a long-standing problem to get them positive and possessing cohomological interpretation. Constructing positive "superpolynomials" is not straightforward, especially for…
We give explicit formulas for the adjoint twisted Alexander polynomial and the nonabelian Reidemeister torsion of genus one two-bridge knots.
In this note, we compute the cyclotomic expansion formula for colored Jones polynomial of double twist knots with an odd number of half-twists $\mathcal{K}_{p,\frac{s}{2}}$ by using the Kauffman bracket skein theory. It answers a question…
We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups…
The colored Jones polynomial is a knot invariant that plays a central role in low dimensional topology. We give a simple and an efficient algorithm to compute the colored Jones polynomial of any knot. Our algorithm utilizes the walks along…
We prove that for any zero {\alpha} of the Alexander polynomial of a two-bridge knot, -3 < Re({\alpha}) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re({\alpha}).
We compute the Ozsvath-Szabo Floer homologies HF^{+-} and HF-hat for three-manifolds obtained by integer surgery on a two-bridge knot.
We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals…
We give an alternative proof to Agol's classification of parabolic generating pairs of non-free Kleinian groups generated by two parabolic transformations. As an application, we give a complete characterisation of epimorphims between…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371…
The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also…