Related papers: On Alexander-Conway polynomials of two-bridge link…
The Vol-Det Conjecture, formulated by Champanerkar, Kofman and Purcell, states that there exists a specific inequality connecting the hyperbolic volume of an alternating link and its determinant. Among the classes of links for which this…
The Alexander polynomials \Delta_{n,3}(t) and \Delta_{n,4}(t) are presented as a sum of the Alexander polynomials \Delta_{k,2}(t). These polynomials are also expressed in the form of a sum of Chebyshev polynomials of the second kind. These…
Suppose the knot group G(K) of a knot K has a non-abelian representation \rho on A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to \rho is of the form: \Delta_K(t)/(1-t) \phi(t^3), where \Delta_K…
We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We…
Motivated by the conjectured asymptotics of the Kashaev invariant, Dimofte and the first author introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic 3-manifold, proved that its constant (1-loop) term…
In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices,…
The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for…
The altenating knots, links and twists projected on the S_2 sphere are identified with the phase Space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossing points, the edges, to the…
We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the…
One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant…
In a recent paper, Allen and Swenberg investigated which link polynomials are capable of detecting causality in (2+1)-dimensional globally hyperbolic spacetimes. They ultimately suggested it is likely that the Jones Polynomial accomplishes…
If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not…
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect…
We extend the state models for Jones and Alexander polynomials of classical links to state models of 2-variable polynomials in the case of singular links. Moreover, we extend both of them to polynomials with d+1 variables for long singular…
We compute the reduced version of Khovanov and Rozansky's sl(N) homology for two-bridge knots and links. The answer is expressed in terms of the HOMFLY polynomial and signature.
We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of…
According to work of Hartley and Kawauchi in 1979 and 1980, the Conway Polynomial of all negative amphicheiral knots and strongly positive amphicheiral knots factors as $\phi(z)\phi(-z)$ for some $\phi(z)\in\mathbb Z[z]$. Moreover, a 2012…
In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.
For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this…
We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the…