The multi-variable Affine Index Polynomial for tangles

TL;DR

This paper extends the multi-variable affine index polynomial to virtual tangles, ensuring compatibility with tangle decomposition, and introduces Turaev moves for virtual tangles, linking crossing weights to homology intersection pairings.

Contribution

It generalizes the affine index polynomial to virtual tangles and introduces Turaev moves, connecting crossing weights with homology intersection pairings.

Findings

The polynomial is compatible with tangle decomposition.

Turaev moves are defined for virtual tangles.

Crossing weights are recoverable via homology intersection pairing.

Abstract

The multi-variable affine index polynomial was defined by the author in previous work. The aim of this short note is to update the definition so it is generalizable to virtual tangles and to show it is compatible with tangle decomposition. We also introduce the Turaev moves for virtual tangles, and discuss how to recover the weight of each crossing as an intersection pairing of homology classes.