A study on combinatorics of link projections: regular projections of the link L6N1 and prolificity of arrangements of pseudocircles

Abstract

The relationship between a link embedded in the 3-sphere and its planar projection (or shadow) is a central subject in knot theory. This thesis investigates two distinct facets of this relationship. First, we address the characterization problem: for a particular link L, a fundamental problem consists of characterizing the exact set of shadows that are projections of L. While this question has been resolved for links with small crossing numbers, we extend this inquiry to the 3-component prime link L6n1. This work provides a complete characterization: any 3-component link projection is in fact a projection of L6n1 under the necessary and sufficient condition that its components intersect pairwise. This is achieved through the identification and analysis of reduction operations that simplify projections while preserving key properties, ultimately characterizing the irreducible projections of L6n1. Second, we explore the prolificity of shadows: what is the total number of distinct links that can be obtained from a particular shadow? We focus specifically on shadows formed by arrangements of pseudocircles (collections of Jordan curves that pairwise intersect exactly twice) and restrict our analysis to the realization of positive oriented links. We establish precise asymptotic bounds for the number of distinct positive oriented links that can be obtained over the three unavoidable classes of pseudocircle arrangements: the ring Rn, the boot Bn, and the flower Fn. We demonstrate that the number of such links, relative to the total number of possible positive realizations (2n), behaves asymptotically as 1/4 for Rn, 1 for Bn, and 1/ (2n) for Fn.