Construction and decomposition of knots as Murasugi sums of Seifert surfaces

TL;DR

This paper explores how knots can be constructed and decomposed using Murasugi sums of Seifert surfaces, revealing the structure of the space of knots and providing bounds on the complexity of such sums.

Contribution

It demonstrates that any knot can be expressed as a Murasugi sum of two knots and analyzes the minimal complexity needed for such decompositions.

Findings

Any knot is a Murasugi sum of any two knots.

The structure of the knot space forms a bi-directed complete graph.

Bounds for minimal Murasugi sum complexity to obtain a third knot.

Abstract

A fixed knot $K$ acts via Murasugi sum on the space $\mathcal{S}$ of isotopy classes of knots. This operation endows $\mathcal{S}$ with a directed graph structure denoted by $M\kern-1pt SG(K)$. We show that any given family of knots in $M\kern-1pt SG(K)$ has the structure of a bi-directed complete graph, which is not the case if we restrict the complexity of Murasugi sums. For that purpose, we show that any knot is a Murasugi sum of any two knots, and we give lower and upper bounds for the minimal complexity of Murasugi sum to obtain $K_3$ by $K_1$ and $K_2$. As an application, we show that given any three knots, there is a braid for one knot which splits along a string into braids for the other two knots.