Circle arrangements of link projections

TL;DR

This paper studies the circle arrangements derived from link projections, characterizing when two projections share the same arrangement and constructing projections with specific arrangements for odd and even component links.

Contribution

It provides a characterization of link projections with identical circle arrangements and constructs projections with minimal circle arrangements for odd and even component links.

Findings

Two link projections have the same circle arrangement iff they are related by certain local moves.

Every odd component link has a projection with one circle in its arrangement.

Every even component link has a projection with two circles in its arrangement.

Abstract

An oriented link projection is the image of a generic immersion of oriented circles into the 2-sphere. The circle arrangement of a link projection is a disjoint union of unoriented circles on the 2-sphere obtained by orientation-incoherent smoothing at each crossing point. We show that two oriented link projections have the same circle arrangement if and only if they are transformed into each other by certain local moves. We also show that every odd (resp. even) component link has a link projection whose circle arrangement consists of exactly one (resp. two) circles.