Knotting and linking in the Petersen family

TL;DR

This paper explores the relationship between knotting and linking in Petersen family graphs, establishing algebraic connections and explicit formulas for specific graphs like K_{3,3,1}.

Contribution

It introduces a novel relationship linking algebraic linking and knotting in Petersen family graphs, including explicit formulas for K_{3,3,1}.

Findings

Established a relationship between algebraic linking and knotting.

Derived explicit formulas connecting linking numbers and knot invariants.

Enhanced understanding of intrinsic linking and knotting in minor minimal graphs.

Abstract

This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more explicit relationship for the graph $K_{3,3,1}$ between knotting and linking, which relates the sum of the squares of linking numbers of links in the embedding and the second coefficient of the Conway polynomial of particular cycles in the embedding.