Related papers: Gaps between consecutive untwisting numbers

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the…

We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on…

We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify…

We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsvath and Szabo's obstruction to unknotting number one. We determine the unknotting numbers of 9_10, 9_13, 9_35, 9_38, 10_53, 10_101 and…

Using the techniques on annulus twists, we observe that $6_3$ has infinitely many non-characterizing slopes, which affirmatively answers a question by Baker and Motegi. Furthermore, we prove that the knots $6_2$, $6_3$, $7_6$, $7_7$, $8_1$,…

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

Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We…

Marino's conjecture remains underexplored within the framework of $SO(N )$ string dualities. In this article, we investigated the reformulated invariants of a one-parameter family of knots $\left[ K\right]_p$ derived from tangle surgery on…

A knot K is called n-adjacent to the unknot, if K admits a projection containing n generalized crossings such that changing any m (no larger than n) of them yields a projection of the unknot. We show that a non-trivial satellite knot K is…

We describe a way of encoding a Kauffman state as a set of tuples, similar to a Gauss code. Then we describe a procedure for using these state codes to determine the unoriented genus and crosscap number of any prime alternating knot or…

In this paper, we show a new upper bound of prime gaps, that is the gap between a prime number and its consecutive prime number. We show that the gap between a prime number $p_n$ and its consecutive prime number is not larger than…

The transient number of a knot K, denoted tr(K), is the minimal number of simple arcs that have to be attached to K, in order that K can be homotoped to a trivial knot in a regular neighborhood of the union of K and the arcs. We give a…

A gordian unlink is a finite number of unknots that are not topologically linked, each with prescribed length and thickness, and that cannot be disentangled into the trivial link by an isotopy preserving length and thickness throughout. In…

Gay and Meier asked if a trisection diagram for the Gluck twist on a spun or twist-spun 2-knot in $S^4$ obtained by a certain method is standard. In this paper, we show that the trisection diagram for the Gluck twist on the spun…

We study spectral gaps of cellular differentials for finite cyclic coverings of knot complements. Their asymptotics can be expressed in terms of irrationality exponents associated with ratios of logarithms of algebraic numbers determined by…

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…

For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K): c(K) \leq [(g(K)+9)/6] and c(K) \leq [(n(K) + 16)/12]. The (6n-2,3) torus knots show that these bounds are sharp.

Batson's conjecture is a non-orientable version of Milnor's conjecture, which states that the 4-ball genus of a torus knot $T(p,q)$ is equal to $\frac{(p-1)(q-1)}{2}$. Batson's conjecture states that the nonorientable 4-ball genus is equal…

We study a local twist move on welded knots that is an analog of the virtualization move on virtual knots. Since this move is an unknotting operation we define an invariant, unknotting twist number, for welded knots. We relate the…