On the Uniqueness of Sarkar-Seed-Szabo Construction

TL;DR

This paper proves the uniqueness of the Sarkar-Seed-Szabo total complex construction, confirming that the added differential terms are essentially unique under certain properties, thereby establishing the invariance of the complex.

Contribution

It demonstrates the uniqueness of the differential terms in Sarkar-Seed-Szabo's total complex, ensuring the construction's well-defined nature.

Findings

The differential terms are unique up to specified properties.

The total complex $CTot(L)$ is uniquely determined by the construction.

The result consolidates different chain complex approaches in knot theory.

Abstract

In an attempt to consolidate Szabo's geometric chain complex and Bar\hyp Natan's chain complex, Sarkar-Seed-Szabo defined a total complex $CTot(L)$ over $\mathbb{F}_{2}$ by adding some extra terms $\{h_{i} \}_{i=2}^{\infty}$ in the differential. In this paper we prove the uniqueness of $\{h_{i} \}_{i=2}^{\infty}$ upto some properties. This in turn implies that the total complex $CTot(L)$ is unique.