On the classification of periodic weaves and universal cover of links in thickened surfaces

TL;DR

This paper introduces hyperbolic periodic weaves, extends classical link conjectures to weaving motifs, and proves these conjectures using a generalized polynomial, advancing the classification of periodic weaves in thickened surfaces.

Contribution

It generalizes the concept of doubly periodic weaves to hyperbolic cases and extends classical link conjectures to weaving motifs, providing new tools for their classification.

Findings

Extended Tait Conjectures to weaving motifs.

Proved minimal crossing and writhe properties for hyperbolic weaves.

Developed a generalized Kauffman bracket polynomial for periodic weaving diagrams.

Abstract

A periodic weave is the lift of a particular link embedded in a thickened surface to the universal cover. Its components are infinite unknotted simple open curves that can be partitioned in at least two distinct sets of threads. The classification of periodic weaves can be reduced to the one of their generating cells, namely their weaving motifs. However, this classification cannot be achieved through the classical theory of links in thickened surfaces since periodicity in the universal cover is not encoded. In this paper, we first introduce the notion of hyperbolic periodic weaves, which generalizes our doubly periodic weaves embedded in the Euclidean thickened plane. Then, Tait First and Second Conjectures are extended to minimal reduced alternating weaving motifs and proved using a generalized Kauffman bracket polynomial defined for periodic weaving diagrams of the Euclidean plane and generalized to the hyperbolic plane. The first conjecture states that any minimal alternating reduced weaving motif has the minimum possible number of crossings, while the second one formulates that two such oriented weaving motifs have the same writhe.