相关论文: Splitting numbers of links and the four-genus

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering…

The splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split link. We obtain a lower bound on the splitting number in terms of the (multivariable) signature and…

We provide an algorithm to determine whether a link L admits a crossing change that turns it into a split link, under some fairly mild hypotheses on L. The algorithm also provides a complete list of all such crossing changes. It can…

The weak splitting number $wsp(L)$ of a link $L$ is the minimal number of crossing changes needed to turn $L$ into a split union of knots. We describe conditions under which certain $\mathbb{R}$-valued link invariants give lower bounds on…

It is known that algebraically split links (links with vanishing pairwise linking number) can be transformed into the trivial link by a series of local moves on the link diagram called delta-moves; we define the delta-unlinking number to be…

A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a…

In this paper, we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete…

The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many…

The slicing degree of a knot $K$ is defined as the smallest integer $k$ such that $K$ is $k$-slice in $\#^n \overline{\mathbb{CP}^2}$ for some $n$. In this paper, we establish bounds for the slicing degrees of knots using Rasmussen's…

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. The algebraic unknotting number is the minimum number of crossing changes needed to transform a knot into an Alexander…

An open question asks if every knot of 4-genus g_s can be changed into a slice knot by g_s crossing changes. A counterexample is given.

We investigate the minimal number of links and knots in complete partite graphs. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices…

The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by…

In links with two components there are three different types of crossings: self-crossings in the first component, self crossings in the second component, and crossings between components. In this paper we examine the minimum number of…

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger…

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove…

In this paper, we enumerate the number of oriented rational knots and the number of oriented rational links with any given crossing number and minimum genus. This allows us to obtain a precise formula for the average minimal genus of…

The twisting number of a ribbon knot $K$ is the minimal number of tangle replacements on the symmetry axis of $J \# -J$ for any knot $J$ that is required to produce a symmetric union diagram of $K$. We prove that the twisting number is…

We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. The main theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing…

The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in…