Computation of the knot symmetric quandle and its application to the plat index of surface-links

TL;DR

This paper computes the knot symmetric quandle for surface-links using plat form presentations and demonstrates the existence of infinitely many surface-knots with specified genus and plat index.

Contribution

It introduces a method to compute the knot symmetric quandle from plat form presentations of surface-links and applies this to show the diversity of surface-knots with given genus and plat index.

Findings

Computed the knot symmetric quandle for surface-links.

Proved the existence of infinitely many surface-knots with fixed genus and plat index.

Abstract

A surface-link is a closed surface embedded in the 4-space, possibly disconnected or non-orientable. Every surface-link can be presented by the plat closure of a braided surface, which we call a plat form presentation. The knot symmetric quandle of a surface-link $F$ is a pair of a quandle and a good involution determined from $F$. In this paper, we compute the knot symmetric quandle for surface-links using a plat form presentation. As an application, we show that for any integers $g \geq 0$ and $m \geq 2$, there exists infinitely many distinct surface-knots of genus $g$ whose plat indices are $m$.