Knot theory and cluster algebra III: Posets

TL;DR

This paper explores the structure of posets associated with link diagrams, revealing they form distributive lattices and establishing connections with coefficient quivers, advancing the understanding of algebraic and combinatorial properties in knot theory.

Contribution

It demonstrates that the posets are distributive lattices, describes join irreducibles explicitly, and links Kauffman states with coefficient quivers, providing new algebraic insights.

Findings

Posets are distributive lattices.

Explicit descriptions of join irreducibles.

Isomorphism between join irreducible Kauffman states and coefficient quivers.

Abstract

In previous work, we associated a module $T(i)$ to every segment $i$ of a link diagram $K$ and showed that there is a poset isomorphism between the submodules of $T(i)$ and the Kauffman states of $K$ relative to $i$. In this paper, we show that the posets are distributive lattices and give explicit descriptions of the join irreducibles in both posets. We also prove that the subposet of join irreducible Kauffman states is isomorphic to the poset of the coefficient quiver of $T(i)$.