Full Colored HOMFLYPT Invariants, Composite Invariants and Congruent Skein Relation

TL;DR

This paper studies the properties of full colored HOMFLYPT invariants within skein theory, introduces composite invariants and a reformulated invariant, and proves their algebraic integrality and a conjectured skein relation.

Contribution

It introduces a new reformulated composite invariant and proves its algebraic integrality, along with proposing and verifying a congruent skein relation.

Findings

Full colored HOMFLYPT invariants have a structured behavior as q approaches 1.

The reformulated composite invariant lies in a specific algebraic ring.

A conjecture on a congruent skein relation is proposed and verified in special cases.

Abstract

In this paper, we investigate the properties of the full colored HOMFLYPT invariants in the full skein of the annulus $\mathcal{C}$. We show that the full colored HOMFLYPT invariant has a nice structure when $q\rightarrow 1$. The composite invariant is a combination of the full colored HOMFLYPT invariants. In order to study the framed LMOV type conjecture for composite invariants, we introduce the framed reformulated composite invariant $\check{\mathcal{R}}_{p}(\mathcal{L})$. By using the HOMFLY skein theory, we prove that $\check{\mathcal{R}}_{p}(\mathcal{L})$ lies in the ring $2\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Furthermore, we propose a conjecture of congruent skein relation for $\check{\mathcal{R}}_{p}(\mathcal{L})$ and prove it for certain special cases.