Generalization of Polynomial Invariants and Holographic Principle for Knots and Links

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.