Related papers: Twisted Alexander polynomials and character variet…

Given a homomorphism from a link group to a group, we introduce a $K_1$-class in another way, which is a generalization of the 1-variable Alexander polynomial. We compare the $K_1$-class with $K_1$-classes in \cite{Nos} and with…

Let K be a knot in $S^3$ and $X$ its complement. We study deformations of reducible metabelian representations of the knot group $\pi_1(X)$ into $SL(3,\mathbb{C})$ which are associated to a double root of the Alexander polynomial. We prove…

C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Giller's invariant is, however, a…

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat…

This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.

We show that the $(4,5)$- and $(5,6)$-torus knots admit ghost characters. Consequently, these knots provide counterexamples to Ng's conjecture, which proposes an isomorphism between the complexification of degree $0$ abelian knot contact…

In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.

We establish a connection between the Alexander polynomial of a knot and its twisted and $L^2$-versions with the triangulations that appear in 3-dimensional hyperbolic geometry. Specifically, we introduce twisted Neumann--Zagier matrices of…

We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise…

The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it…

A polynomial f(t) with rational coefficients is strongly irreducible if f(t^k) is irreducible for all positive integers k. Likewise, two polynomials f and g are strongly coprime if f(t^k) and g(t^l) are relatively prime for all positive…

We show that all twist knots, certain double twist knots and some other 2-bridge knots are minimal elements for the partial ordering on the set of prime knots. The key to these results are presentations of their character varieties using…

We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties.…

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume…

We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how…

We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.

In this paper, we extend the symplectic quandle method, previously employed in our study of parabolic representations of knot groups, to investigate the general $SL(2,\mathbb{C})$-representations of 2-bridge ``kmot" groups. We introduce a…

Let M be a two cusped hyperbolic 3-manifold and let M(r) be the result of r Dehn filling of a fixed cusp of M. We study canonical components of the SL(2,C) character varieties of M(r). We show that the gonality of these sets is bounded,…

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…

In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely…