arXiv research

New Garside structures found for torus knot groups and related braid groups.

Characterizes character varieties of generalized torus knot groups.

Formula for Alexander polynomial of twisted torus knots derived.

This paper extends knot invariants using instantons to study torus knot groups.

Study calculates fundamental groups of torus knots using algebraic topology.

It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves…

The paper explores non-trivial welded knots and ribbon torus-knots, proving their existence.

New method detects sliceness of knots in a torus.

New infinite class of hyperbolic knots with high genus and generalized torsion found.

We show that for many classical knots one can find generalized torsion in the fundamental group of its complement, commonly called the knot group. It follows that such a group is not bi-orderable. Examples include all torus knots, the (hyperbolic) knot $5_2$ and satellites of these knots.

Study spaces of knots and links in specific 3-manifolds.

New reflection groups derived from torus knots with finite meridians.

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none the automorphic image of another, such that each normally generates the group.

Paper develops knot invariants for long knots in a torus.

A torus-covering $T^2$-knot is a surface-knot of genus one determined from a pair of commutative braids. For a torus-covering $T^2$-knot $F$, we determine the number of irreducible metabelian $SU(2)$-representations of the knot group of $F$ in terms of the knot determinant of $F$. It is similar to the result due to Lin…

New invariant defined for unoriented knots, proving no factorization through topological concordance.

We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG(T,2) with a standard element only depending on …

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable …

Study satellite knots and their quandles related to incompressible tori.

Boyer, Gordon, and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since large classes of L-spaces can be produced from Dehn surgery on knots in the 3-sphere, it is natural to ask what conditions on the knot group are suffi…

Harer-Zagier formulas generalized to knot matrix models.

This paper's theme is the relation between several classical and well-known objects: triangle Fuchsian groups, quasi-homogeneous singularities of plane curves, torus knot complements in the 3-sphere. Torus knots are the only nontrivial knots whose complements admit transitive Lie group actions. In fact S^3\K_{p,q} is d…

The paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

New examples show satellite operations can expand the concordance group in topological knot theory.

For an arbitrary positive integer $n$ and a pair $(p, q)$ of coprime integers, consider $n$ copies of a torus $(p,q)$ knot placed parallel to each other on the surface of the corresponding auxiliary torus: we call this assembly a torus $n$-link. We compute economical presentations of knot groups for torus links using t…

Method produces faithful representations of Garside groups and torus knot groups.

Study on quantum invariants from surgeries on torus knots.

Study cobordism distances between 3-braid links and trefoil knots.

We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non left-orderable fundamental groups.

The paper studies random covers of torus knot complements and their statistical properties.

Lueck expressed the Gromov norm of a knot complement in terms of an infinite series that can be computed from a presentation of the fundamental group of the knot complement. In this note we show that Lueck's formula, applied to torus knots, yields surprising power series expansions for the logarithm function. This gene…

Detects torus knots using SL(2,C) representations and instanton Floer homology.

We consider the relations $\ge$ and $\ge_p$ on the collection of all knots, where $k \ge k'$ (respectively, $k \ge_p k'$) if there exists an epimorphism $πk \to πk'$ of knot groups (respectively, preserving peripheral systems). When $k$ is a torus knot, the relations coincide and $k'$ must also be a torus knot; we dete…

Defines a measure of knot concordance using cobordism distance.

Motivated by Clay and Watson's question on left-orderability of the fundamental group of the resultant space of an $r'$-surgery on the $(p, q)$-cable knots for $r' \in (pq-p-q,pq)$, this paper proves by elementary means that for specific pairs of $(p,q)$-cable knots of torus knots, $r' \in [pq-1,pq]$ gives a surgery yi…

In this paper we give an introduction to the volume conjecture and its generalizations. Especially we discuss relations of the asymptotic behaviors of the colored Jones polynomials of a knot with different parameters to representations of the fundamental group of the knot complement at the special linear group over com…

Study SO(3)-knot states for torus complements, linking to simplicial volume.

In this note, we investigate genera for the slopes of a knotted torus in the 4-sphere analogous to the genus of a classical knot. We compare various formulations of this notion, and use this notion to study the extendable subgroup of the mapping class group of the knotted torus.

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 and the Conway's normal form for 2-bridge knots. For a given Schubert's normal f…

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 A-polynomial of the knot. For example, we show that for 2-bridge knots the polynomial…

New slopes identified for torus knots, improving previous results.

We show that a knot has a non left-orderable surgery if the knot group admits a generalized Baumslag-Solitar relator and satisfies certain conditions on a longitude of the knot. As an application, it is shown that certain positively twisted torus knots admit non left-orderable surgeries.

Study the geometry of torus link character varieties, finding unexpected relations.

Lee's work on twisted torus knots with Fibonacci parameters is extended to Horadam parameters.

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 Reidemeister torsions associated with various representations appear in the asymptotic expa…

We compute the Chekanov-Eliashberg contact homology of what we call the Legendrian closure of a positive braid. We also construct an augmentation for each such link diagram. Then we apply the monodromy techniques established in an earlier paper to a certain natural loop in the space L' of positive Legendrian (p,q) toru…

We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal-Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of log |the high…

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern--Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers.…