arXiv research

Study proves bridge and braid indices match for twist positive knots.

Positive braids with at least two twists form hyperbolic knots.

The study connects twist positivity to L-space knots and concordance.

New satellite knots found that can't be represented by positive braids with full twists.

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 use Ozsváth, Stipsicz, and Szabó's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality for knots: positive braids that contain a positive full-twist realize the braid inde…

Quandle cocycle invariants form a powerful and well developed tool in knot theory. This paper treats their variations - namely, positive and twisted quandle cocycle invariants, and shadow invariants. We interpret the former as particular cases of the latter. As an application, several constructions from the shadow worl…

Let $\mathcal{L}$ be a knot with a fixed positive crossing and $\mathcal{L}_n$ the link obtained by replacing this crossing with $n$ positive twists. We prove that the knot Floer homology $\widehat{\text{HFK}}(\mathcal{L}_n)$ `stabilizes' as $n$ goes to infinity. This categorifies a similar stabilization phenomenon of …

Paper calculates the ribbonlength of twisted torus knots.

Study of knot invariant growth for twisted knots.

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

The study finds infinitely many twist knot complements with totally geodesic surfaces.

We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), when the substitute of the strand was a torus knot. We describe a general decomposition of satellite's colored HOMFLY in …

The paper shows knots can have zero support genus in certain spaces.

The paper finds left orderable slopes for specific double twist knots.

The paper constructs triangulations for double twist knots using geometric methods.

Highly twisted knots can be geometrically triangulated.

Conjecturally, there are only finitely many Heegaard Floer L-space knots in $S^3$ of a given genus. We examine this conjecture for twist families of knots $\{K_n\}$ obtained by twisting a knot $K$ in $S^3$ along an unknot $c$ in terms of the linking number $ω$ between $K$ and $c$. We establish the conjecture in case of…

We find an infinite family of Seifert fibered surgeries on strongly invertible knots which do not have primitive/Seifert positions. Each member of the family is obtained from a trefoil knot after alternate twists along a pair of seiferters for a Seifert fibered surgery on a trefoil knot.

We develop a model characterizing all possible knots and links arising from recombination starting with a twist knot substrate, extending previous work of Buck and Flapan. We show that all knot or link products fall into three well-understood families of knots and links, and prove that given a positive integer $n$, the…

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-…

Characterizes unknotted curves on Seifert surfaces of twist knots.

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 integers k and l. We provide some sufficient conditions for strong irreducibility a…

The article shows how to create metrics with positive Ricci curvature on twisted suspensions.

We show, using Mellit's recent results, that Kálmán's full twist formula for the HOMFLY polynomial can be generalized to a formula for superpolynomials in the case of positive toric braids.

Satellite links of fully positive braids are characterized.

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.

New infinite families of twisted torus knots found.

We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the negatively twisted, positive Whitehead doubles of all knots. We also verify the conjectur…

Hom and Wu introduced a knot concordance invariant called $ν^+$, which dominates many concordance invariants derived from Heegaard Floer homology. In this paper, we give a full-twist inequality for $ν^+$. By using the inequality, we extend Wu's cabling formula for $ν^+$ (which is proved only for particular positive cab…

The paper studies hyperbolic geometry of links formed by adding trivial components to knots.

Classifies 3-braid knots with maximal 4-genus using McCoy's method.

Formula for Alexander polynomial of twisted torus knots derived.

We create Lefschetz fibrations for knot traces of specific types of knots.

We prove that many pretzel knots of the form $P(2n,m,-2n\pm1,-m)$ are not topologically slice, even though their positive mutants $P(2n, -2n\pm1, m, -m)$ are ribbon. We use the sliceness obstruction of Kirk and Livingston related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.

Paper shows how to twist knots to make them trivial or non-trivial.

New families of twisted torus knots found with essential surfaces.

Study lattice paths from twist knots and double twist knots.

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

Study finds infinite non-fibered twisted torus knots.

Study on singular twisted links and virtual braids, extending knot theory concepts.

The paper extends knot theory to twisted virtual braids and links.

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

Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prov…

Knot Floer homology stabilizes with twists.

Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot (i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery), and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently positi…

Proved colored HOMFLY-PT polynomials for specific knots.

The study explores knots and manifolds, proving properties and non-left-orderable groups.