Seifert circles, crossing number and the braid index of generalized knots and links

TL;DR

This paper extends inequalities relating crossing number and braid index from classical and virtual links to generalized knots and links, including virtual singular links, using new concepts of generalized crossings and Reidemeister moves.

Contribution

It introduces generalized crossings and Reidemeister moves to study links independently of crossing types, proving analogous inequalities for a broader class of links.

Findings

Inequality between total crossing number and braid index holds for generalized knots.

Results apply to virtual singular links.

Framework unifies classical, virtual, and singular links.

Abstract

For classical links Ohyama proved an inequality involving the minimal crossing number and the braid index, then motivated from this Takeda showed an analogous inequality for virtual links. In this paper, we are interested in studying properties of links independent of the type of crossings, and for this reason, we introduce generalized crossings for diagrams and generalized Reidemeister-type moves. The aim of this work is to prove the same type of inequality mentioned above but now involving the total crossing number and the braid index of generalized knots and links. In particular, we show that the result holds for virtual singular links.