Related papers: On the algebraic unknotting number

A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound…

We define the stabilizing number $\operatorname{sn}(K)$ of a knot $K \subset S^3$ as the minimal number $n$ of $S^2 \times S^2$ connected summands required for $K$ to bound a nullhomotopic locally flat disc in $D^4 \# n S^2 \times S^2$.…

In this paper we introduce a new invariant of virtual knots and links that is non-trivial for infinitely many virtuals, but is trivial on classical knots and links. The invariant is initially be expressed in terms of a relative of the…

Ozsvath and Szabo have defined a knot concordance invariant tau that bounds the 4-ball genus of a knot. Here we discuss shortcuts to its computation. We include examples of Alexander polynomial one knots for which the invariant is…

The genus non-increasing totally positive unknotting number is the minimum number of crossing changes that transform a knot into the unknot, such that all the crossing changes are positive-to-negative crossing changes that do not increase…

The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.

The number $|K|$ of non-isotopic framed knots that correspond to a given unframed knot $K\subset S^3$ is infinite. This follows from the existence of the self-linking number $\slk$ of a zerohomologous framed knot. We use the approach of…

In this paper we introduce the notion of an unknotting index for virtual knots. We give some examples of computation by using writhe invariants, and discuss a relationship between the unknotting index and the virtual knot module. In…

The Links-Gould invariant of links $LG^{2,1}$ is a two-variable generalization of the Alexander-Conway polynomial. Using representation theory of $U_{q}\mathfrak{gl}(2 \vert 1)$, we prove that the degree of the Links-Gould polynomial…

Based on a vanishing theorem for non-fibered knots due to Friedl and Vidussi, we define the twisted Alexander vanishing order of a knot to be the order of the smallest finite group such that the corresponding twisted Alexander polynomial is…

We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture…

The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and…

Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. I will present my multivariate version of Bigelow's calculation. The advantage to my…

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…

In 1991, Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$ of the knot which is $s(K) \leq 2 c(K)$. In this paper we improve this upper bound to $s(K) \leq…

We showed that the order of torsion homology classes in the grid homology of a knot is a lower bound for the unknotting number.

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…

The unoriented band unknotting number of a knot is the minimum number of oriented or non-oriented band surgeries that turn the knot into the unknot. Batson introduced a certain non-oriented band surgery for a torus knot. The minimum number…

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard,…

We present a sequence of diagrams of the unknot for which the minimum number of Reidemeister moves required to pass to the trivial diagram is quadratic with respect to the number of crossings. These bounds apply both in $S^2$ and in $\R^2$.