Full Colored HOMFLYPT Invariants, Composite Invariants and Congruent Skein Relation

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 $q\to 1$. 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.

arXiv comment27 pages, 3 figures

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