HOMFLY Polynomial Invariants of Torus Knots and Bosonic (q,p)-Calculus

TL;DR

This paper introduces bosonic (q,p)-numbers linked to deformed bosonic oscillators to derive HOMFLY polynomial invariants of torus knots, connecting one-parameter Alexander and Jones skein relations with the two-parameter HOMFLY skein relation.

Contribution

It proposes a novel bosonic (q,p)-number framework that unifies and derives HOMFLY polynomial invariants from simpler skein relations.

Findings

HOMFLY skein relation can be obtained from Alexander and Jones relations using bosonic (q,p)-numbers.

Two equivalent methods to derive HOMFLY invariants from one-parameter skein relations.

Bosonic (q,p)-numbers connect knot invariants with deformed bosonic oscillators.

Abstract

For the one-parameter Alexander (Jones) skein relation we introduce the Alexander (Jones) "bosonic" q-numbers, and for the two-parameter HOMFLY skein relation we propose the HOMFLY "bosonic" (q,p)-numbers ("bosonic" numbers connected with deformed bosonic oscillators). With the help of these deformed "bosonic" numbers, the corresponding skein relations can be reproduced. Analyzing the introduced "bosonic" numbers, we point out two ways of obtaining the two-parameter HOMFLY skein relation ("bosonic" (q,p)-numbers) from the one-parameter Alexander and Jones skein relations (from the corresponding "bosonic" q-numbers). These two ways of obtaining the HOMFLY skein relation are equivalent.