arXiv research

The study finds hyperbolic twisted torus links for certain twists.

The paper classifies twisted torus links that are unlinks.

New method calculates number of components in twisted torus links.

Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their …

New findings on T-links derived from torus links.

Jones polynomial for twisted torus knots is trivial if and only if the knot is trivial.

Study Alexander polynomials of links in 3-torus.

The paper classifies hyperbolic and satellite T-links formed by twisting.

We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…

In this paper we give an explicit formula for the twisted Alexander polynomial of any torus link and show that it is a locally constant function on the $SL(2, \mathbb C)$-character variety. We also discuss similar things for the higher dimensional twisted Alexander polynomial and the Reidemeister torsion.

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…

Classifies small links in an unmarked solid torus.

Generalizes Hecke algebra for double torus, linking to skein algebra.

We present a simple combinatorial model for quasipositive surfaces and positive braids, based on embedded bipartite graphs. As a first application, we extend the well-known duality on standard diagrams of torus links to twisted torus links. We then introduce a combinatorial notion of adjacency for bipartite graph links…

Positive braids with at least two twists form hyperbolic knots.

A Coxeter link is a closure of a product of two braids, one being a quasi-Coxeter element and the other being a product of partial full twists. This class of links includes torus knots \(T_{n,k}\) and torus links \(T_{n,nk}\). We identify the knot homology of a Coxeter link with the space of sections of a particular li…

In this paper, we study the Khovanov homology of cable links. We first estimate the maximal homological degree term of the Khovanov homology of the ($2k+1$, $(2k+1)n$)-torus link and give a lower bound of its homological thickness. Specifically, we show that the homological thickness of the ($2k+1$, $(2k+1)n$)-torus li…

Study on the ribbonlength of knots and links, improving upper bounds.

The paper calculates colored Jones polynomials for specific link configurations.

New infinite families of twisted torus knots found.

Knot invariants and quiver stability linked through full twists.

The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.

HZ transform applied to knot polynomials reveals hyperbolic knot structures.

Formula for Alexander polynomial of twisted torus knots derived.

New families of twisted torus knots found with essential surfaces.

This paper is a continuation on the 2012 paper on "Cutting Twisted Solid Tori (TSTs)", in which we considered twisted solid torus links (tst links). We generalize the notion of tst links to "surgerized tst links": recall that when performing $Φ^μ(n(τ), d(τ), M)$ on a tst $\langle τ\rangle$ where $M$ is odd, we obtain t…

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

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in a particular way. Dean gives sufficient conditions for the parameters of the tw…

We construct complexes $P_{1^n}$ of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for $P_{1^n}$ an…

Study finds infinite non-fibered twisted torus knots.

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can…

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$ if $n<2τ(K_2)$ holds. We also give a formula of Ozsváth-Szabó's $τ$-invariant as…

We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other th…

Computes A-polynomials of knots from Whitehead sister link fillings.

Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of crossing exchanges and smoothings on $D$ that yields a diagram of $L$? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if $L$ is a prime link with crossing number at …

We give a necessary condition for a torus knot to be untied by a single twisting. By using this result, we give infinitely many torus knots that cannot be untied by a single twisting.

The twisted torus knots lie on the standard genus 2 Heegaard surface for $S^3$, as do the primitive/primitive and primitive/Seifert knots. It is known that primitive/primitive knots are fibered, and that not all primitive/Seifert knots are fibered. Since there is a wealth of primitive/Seifert knots that are twisted tor…

Paper calculates the ribbonlength of twisted torus knots.

The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological entropy is the logarithmic growth rate of the number of torsion elem…

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In o…

Homology of the circle with non-trivial local coefficients is trivial. From this well-known fact we deduce geometric corollaries concerning links of codimension two. In particular, the Murasugi-Tristram signatures are extended to invariants of links formed of arbitrary oriented closed codimension two submanifolds of an…

In the present note, we will show that there are infinitely many composite twisted torus knots.

New definition of twisted 1-loop invariant using Ptolemy coordinates.

We show that twisted torus knots $T(p,q,3,s)$ are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.

Formula found for braid index of $n$-bridge braids.

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

The paper calculates knot invariants using Blanchfield forms and obstructs sliceness.

The paper introduces colorings and invariants for twisted links and shows how double coverings can be equivalent.