A Note on Alternating Links and Root Polytopes

TL;DR

This paper explores the connection between the determinant of alternating links and associated root polytopes, revealing that the number of spanning arborescences equals the link determinant when the graph is properly oriented.

Contribution

It establishes a novel relationship linking link determinants, root polytopes, and spanning arborescences in the context of alternating links.

Findings

Determinant of an alternating link equals the number of spanning arborescences in a properly oriented graph.

The link determinant corresponds to the value at -1 of the Jones polynomial.

A new geometric interpretation of link invariants via root polytopes.

Abstract

In this paper, a relationship between the determinant of an alternating link and a certain polytope obtained from the link diagram is analyzed. We also show that when the underlying graph of the link diagram is properly oriented, the number of its spanning arborescence is equal to the determinant, i.e., the value at -1 of the Jones polynomial.