Related papers: Torus-covering knot groups and their irreducible m…
The unknotting number of a knot is bounded from below by its slice genus. It is a well-known fact that the genera and unknotting numbers of torus knots coincide. In this note we characterize quasipositive knots for which the genus bound is…
We calculate the twisted Reidemeister torsion of the complement of an iterated torus knot associated with a representation of its fundamental group to the complex special linear group of degree two. We also show that the twisted…
We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering $T^2$-link is…
This work is concerned with the calculation of the fundamental group of torus knots. Torus knots are special types of knots which wind around a torus a number of times in the longitudinal and meridional directions. We compute and describe…
We briefly review the instanton-torus knot duality found in 5d SUSY gauge theories with one compact dimension. The fermion condensate turn out to be the generating function for the torus knot polynomials colored by the fundamental…
For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot…
We study the existence of irreducible $SU(2)$-representations for cyclic branched covers of knots in $S^3$. Our main result establishes that if $K$ is a non-trivial prime knot and $d$ is an integer such that $d \geq 2$ and $\Sigma_d(K)$ is…
Let $K\subset S^3$ be a knot, $X:= S^3\setminus K$ its complement, and $\mathbb{T}$ the circle group identified with $\mathbb{R}/\mathbb{Z}$. To any oriented long knot diagram of $K$, we associate a quadratic polynomial in variables…
We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured…
A torus-covering $T^2$-link of degree $n$ is a surface-link consisting of tori, in the form of an unbranched covering of degree $n$ over the standard torus. We focus on a torus-covering $T^2$-link of degree 3, which is determined by a pair…
We show that the problem of recognizing that a knot diagram represents a specific torus knot, or any torus knot at all, is in the complexity class ${\sf NP} \cap {\sf co\text{-}NP}$, assuming the generalized Riemann hypothesis. We also show…
Given a knot K and an irreducible metabelian SL(n,C) representation we establish an equality for the dimension of the first twisted cohomology. In the case of equality, we prove that the representation must have finite image and that it is…
We consider surface links in the 4-space which are presented by the form of simple branched coverings over the standard torus, which we call torus-covering links. In this paper, we study unknotting numbers of torus-covering links. In some…
We give a description of all (1,2)-knots in S^3 which admit a closed meridionally incompressible surface of genus 2 in their complement. That is, we give several constructions of (1,2)-knots having a meridionally incompressible surface of…
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…
In this paper we will associate a family $\{K_1,\dots,K_l\}\subset \mathbb{S}^3$ of iterated torus knots to a given free numerical semigroup. We will describe the fundamental group of the knot complement of each knot of the family. Finally,…
We show that a torus knot which is not 2-bridge has a unique irreducible bridge splitting of positive genus.
The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle…
We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In…
We use microlocal sheaf theory to show that if two knots have Legendrian isotopic conormal tori, then the knots are isotopic or mirror images.