Complementary Regions of Multi-Crossing Projections of Knots

TL;DR

This paper extends the concept of universal face sequences to multi-crossing knot projections, proving that (1,2,3,4) is universal for all n-crossing projections with n>2 and providing methods to construct such projections.

Contribution

It introduces the notion of multi-crossing projections and proves the universality of the sequence (1,2,3,4) for all n-crossing projections, along with construction techniques.

Findings

(1,2,3,4) is universal for all n-crossing projections with n>2

All knots have n-crossing projections for all positive n

Construction of n-crossing template knots enables multi-crossing projections

Abstract

An increasing sequence of integers is said to be universal for knots if every knot has a reduced regular projection on the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. Adams, Shinjo, and Tanaka have, in a work, shown that (2,4,5) and (3,4,n) (where n is a positive integer greater than 4), among others, are universal. In a forthcoming paper, Adams introduces the notion of a multi-crossing projection of a knot. An n-crossing projection} is a projection of a knot in which each crossing has n strands, rather than 2 strands as in a regular projection. We then extend the notion of universality to such knots. These results allow us to prove that (1,2,3,4) is a universal sequence for both n-crossing knot projections, for all n>2. Adams further proves that all knots have an n-crossing projection for all positive n. Another proof of this fact is included in this paper. This is achieved by constructing n-crossing template knots, which enable us to construct multi-crossing projections with crossings of any multiplicity.