Generalization of Polynomial Invariants and Holographic Principle for Knots and Links

TL;DR

This paper applies the holographic principle to knots and links, deriving skein relations and generalizations of the Jones polynomial using (q,p)-numbers and recurrence relations for torus knots and links.

Contribution

It introduces a holographic framework for understanding polynomial invariants of knots and links, deriving new skein relations and generalizations.

Findings

Derived the Jones skein relation from the holographic principle.

Generalized skein relations using (q,p)-numbers.

Connected polynomial invariants with (q,p)-deformed oscillators.

Abstract

We formulate the holographic principle for knots and links. For the "space" of all knots and links, torus knots T(2m+1,2) and torus links L(2m,2) play the role of the "boundary" of this space. Using the holographic principle, we find the skein relation of knots and links with the help of the recurrence relation for polynomial invariants of torus knots T(2m+1,2) and torus links L(2m,2). As an example of the application of this principle, we derive the Jones skein relation and its generalization with the help of some variants of (q,p)-numbers, related with (q,p)-deformed bosonic oscillators.