Polynomial Invariants of Torus Knots and (p,q)-Calculus
Anatoliy M. Pavlyuk
TL;DR
This paper introduces deformed fermionic numbers linked to skein relations, enabling the reconstruction of knot invariants like Alexander, Jones, and HOMFLY polynomials through new (p,q)-number formulations.
Contribution
It presents novel deformed fermionic and (p,q)-numbers that generalize skein relations for knot invariants, providing a new algebraic framework.
Findings
Defined fermionic numbers for skein relations
Restored skein relations using fermionic numbers
Introduced (p,q)-numbers for HOMFLY invariants
Abstract
We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we introduce corresponding Alexander (Jones) fermionic q-numbers, and for the HOMFLY skein relation - the HOMFLY deformed (p,q)-numbers with one fermionic parameter.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Geometric and Algebraic Topology · Mathematics and Applications