Related papers: Untwisting number and Blanchfield pairings

Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number.…

In this paper, we study the unknotting operation for twisted knots, called arc shift move. First, we find a family of twisted knots with arc shift number $n$ for any given $n \in \mathbb{N}$. Then we define a new unknotting operation,…

We prove that fibred knots cannot be untied with $\bar{t}_{2k}$-moves, for all $k \geq 2$. More generally, we give an upper bound on the number of two strand twist operations that allow to untie a knot with non-trivial HOMFLY polynomial, in…

We study some comparison between a bilinear cohomology pairing in local coefficients and the Blanchfield pairing of a knot. We show that the former pairing is an $S$-equivalent invariant, and give a criterion to a relation between the two…

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…

For $p\geq 1$ one can define a generalization of the unknotting number $tu_p$ called the $p$th untwisting number which counts the number of null-homologous twists on at most $2p$ strands required to convert the knot to the unknot. We show…

We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 8_{16}, 9_{49} and 6…

Noting that cycle diagrams of permutations visually resemble grid diagrams used to depict knots and links in topology, we consider the knot (or link) obtained from the cycle diagram of a permutation. We show that the permutations which…

We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional…

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let $K$ be a genus one strongly invertible slice knot with nontrivial Alexander polynomial. We show…

We show that the knots $K\in\{4_1,5_1\}$ can be paired with a corresponding knot $K^\prime$ such that $u(K\#K^\prime)<u(K)+u(K^\prime)$. As a consequence unknotting number fails to be additive for these knots. We also provide a candidate…

Knotoid theory is a generalization of knot theory introduced by Turaev in 2012. In recent years, various invariants of knotoids have been studied. In this paper, we mainly discuss unknotting moves and unknotting numbers of plus-welded…

We prove a decomposition formula for twisted Blanchfield pairings of 3-manifolds. As an application we show that the twisted Blanchfield pairing of a 3-manifold obtained from a 3-manifold Y with a representation $\phi: Z[\pi_1(Y)] \to R$,…

A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of…

We introduce and study knots and links in 2-dimensional complexes. In particular, we define linking numbers for oriented two-component links in 2-complexes and a Kauffman-type bracket polynomial for links in 2-complexes. We also discuss…

Unknotting numbers for torus knots and links are well known. In this paper, we present a method for determining the position of unknotting number crossing changes in a toric braid B(p, q) such that the closure of the resultant braid is…

Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer…

We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an…

We introduce a new numerical invariant of knots and links from the descending diagrams. It is considered to live between the unknotting number and the bridge number.

We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we…