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2015•Unpublished venueRequires access

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

A.M. Pavlyuk

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Abstract

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

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What this paper is about

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

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Available abstract

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

Key concepts: Skein, Skein relation, HOMFLY polynomial, Mathematics, Torus, Relation (database), Knot theory, Bracket polynomial

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