相关论文: (1,1)-knots via the mapping class group of the twi…
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…
We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1,1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and…
We show that every strongly-cyclic branched covering of a (1,1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds…
Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng.…
Let $G$ be the fundamental group of the complement of the torus knot of type $(m,n)$. This has a presentation $G=<x,y|x^m=y^n>$. We find the geometric description of the character variety $X(G)$ of characters of representations of $G$ into…
Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. We show that if there is a closed meridionally incompressible surface in the…
A 1-bridge torus knot in a 3-manifold of genus $\le 1$ is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form…
A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the…
The aim of this paper is to investigate the relations between Seifert manifolds and (1,1)-knots. In particular, we prove that every orientable Seifert manifold with invariants {Oo,0|-1;(p,q),...,(p,q),(l, l-1)} has a cyclically presented…
In this paper we study the connections between cyclic presentations of groups and branched cyclic coverings of (1,1)-knots. In particular, we prove that every n-fold strongly-cyclic branched covering of a (1,1)-knot admits a cyclic…
Let G be the fundamental group of the complement of the torus knot of type (m,n). This has a presentation G=<x,y|x^m=y^n>. We find the geometric description of the character variety X(G) of characters of representations of G into SL(3,C),…
Let F a closed connected orientable surface bounding a genus g handlebody H. In this paper we find a finite set of generators for the subgroup E(2,g) of the pure mapping class group of the twice punctured torus PMCG(2,g), consisting of the…
We study the invariant of knots in lens spaces defined from quantum Chern-Simons theory. By means of the knot operator formalism, we derive a generalization of the Rosso-Jones formula for torus knots in L(p,1). In the second part of the…
Let G be the fundamental group of the complement of the torus knot of type (m,n). We study the relationship between SU(2) and SL(2,C)-representations of this group, looking at their characters. Using the description of the SL(2,C)-character…
We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying…
We obtain an explicit representation, as Dunwoody manifolds, of all cyclic branched coverings of torus knots of type $(p,mp\pm 1)$, with $p>1$ and $m>0$.
In this paper we study the knot Floer homology of a subfamily of twisted $(p, q)$ torus knots where $q \equiv\pm1$ (mod $p$). Specifically, we classify the knots in this subfamily that admit L-space surgeries. To do calculations, we use the…
We describe which knots can be obtained as cycles in the canonical book representation of K_n, the complete graph on n vertices. We show that the canonical book representation of K_n contains a Hamiltonian cycle that is a composite knot if…
A (1,1) knot K in a 3-manifold M is a knot that intersects each solid torus of a genus 1 Heegaard splitting of M in a single trivial arc. Choi and Ko developed a parameterization of this family of knots by a four-tuple of integers, which…
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,…