A bracket polynomial for graphs. III. Vertex weights

TL;DR

This paper extends the marked-graph bracket polynomial to weighted graphs, providing formulas for efficient computation in graphs with twin vertices or constructed via graph composition, linking these to link diagram construction.

Contribution

It introduces formulas for weighted bracket polynomial computation in graphs with twin vertices and graph composition, connecting graph theory to link diagram construction.

Findings

Formulas simplify weighted bracket polynomial calculations.

Graph composition corresponds to link diagram construction.

Efficient computation methods for specific graph classes.

Abstract

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e., looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.