Isolated regions of a link projection

TL;DR

This paper characterizes link projections with a maximum of one isolated region, provides formulas for generating functions of such sets, and discusses implications for welded unknotting numbers and combinatorial methods.

Contribution

It determines all link projections with isolate-region number one and introduces a combinatorial approach and formulas for analyzing isolated-region sets.

Findings

All link projections with isolate-region number one are classified.

A formula for the generating function of isolated-region sets is provided for (2, n)-torus links.

Estimations for welded unknotting number are discussed.

Abstract

A set of regions of a link projection is said to be isolated if any pair of regions in the set share no crossings. The isolate-region number of a link projection is the maximum value of the cardinality for isolated sets of regions of the link projection. In this paper, all the link projections of isolate-region number one are determined. Also, estimations for welded unknotting number and combinatorial way to find the isolate-region number are discussed, and a formula of the generating function of isolated-region sets is given for the standard projections of $(2, n)$-torus links.