New structures for colored HOMFLY-PT invariants

DOI10.1007/S11425-021-1951-7arXiv2105.02037MaRDI QIDQ6366888

Shengmao Zhu

Publication date: 5 May 2021

Abstract: In this paper, we present several new structures for the colored HOMFLY-PT invariants of framed links. First, we prove the strong integrality property for the normalized colored HOMFLY-PT invariants by purely using the HOMFLY-PT skein theory developed by H. Morton and his collaborators. By this strong integrality property, we immediately obtain several symmetric properties for the full colored HOMFLY-PT invariants of links. Then, we apply our results to refine the mathematical structures appearing in the Labastida-Mari~no-Ooguri-Vafa (LMOV) integrality conjecture for framed links. As another application of the strong integrality, we obtain that the $q=1$ and $a=1$ specializations of the normalized colored HOMFLY-PT invariant are well-defined link polynomials. We find that a conjectural formula for the colored Alexander polynomial which is the $a=1$ specialization of the normalized colored HOMFLY-PT invariant implies that a special case of the LMOV conjecture for frame knot holds.

Mathematics Subject Classification ID

String and superstring theories; other extended objects (e.g., branes) in quantum field theory (81T30) Finite-type and quantum invariants, topological quantum field theories (TQFT) (57K16)

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