Generalized #-unknotting operations

Unknotting Unknots

A knot is 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. In essence, an unknot is a knot that may be deformed to a standard circle without passing through itself. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unkno...

The Generalized Wiener Polarity Index of some Graph Operations

Let G be a simple connected graph. The generalized polarity Wiener index of G is defined as the number of unordered pairs of vertices of G whose distance is k. Some formulas are obtained for computing the generalized polarity Wiener index of the Cartesian product and the tensor product of graphs in this article.

Stable Operations in Generalized Cohomology

Unknotting Genus One Knots

There is no known algorithm for determining whether a knot has unknotting number one, practical or otherwise. Indeed, there are many explicit knots (11328 for example) that are conjectured to have unknotting number two, but for which no proof of this fact is currently available. For many years, the knot 810 was in this class, but a celebrated application of Heegaard Floer homology by Ozsváth an...

Levelling an unknotting tunnel

It is a consequence of theorems of Gordon–Reid [4] and Thompson [8] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [6], who showed that the (now known) classification of unknotting tunnels for 2–bridge knot...