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Current issue   Ukr. J. Phys. 2014, Vol. 58, N 12, p.1178-1181
doi:10.15407/ujpe58.12.1178    Paper

Pavlyuk A.M.

Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
(14b, Metrolohichna Str., Kyiv 03143, Ukraine; e-mail: pavlyuk@bitp.kiev.ua)

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

Section: General problems of theoretical physics
Original Author's Text: English

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 oneparameter Alexander and Jones skein relations (from the corresponding “bosonic” q-numbers). These two ways of obtaining the HOMFLY skein relation are equivalent.

Key words: polynomial invariant; knot; link; Alexander, Jones, and HOMFLY skein relations; “bosonic” q-numbers; “bosonic” (q, p)-numbers.