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1993•Journal of Knot Theory and Its RamificationsRequires access

QUANTUM INVARIANTS GIVEN BY EVALUATION OF KNOT POLYNOMIALS

H. R. Morton

Open publisher page 7 citations

Abstract

It is shown that the knot invariant arising from an irreducible representation of a quantum group is, under certain conditions, an evaluation of the Homfly or Dubrovnik polynomial of the knot. Besides the known cases of the fundamental representation for each of the quantum groups in the series An, Bn, Cn and Dn, the results cover the special cases of the 3-dimensional representation of SU(2) and the 6-dimensional representation of SU(4), which can be viewed as the fundamental representations of SO(3) and SO(6) respectively. The second of these cases leads to a new relation between an evaluation of the Dubrovnik polynomial of a knot and an evaluation of the Homfly polynomials of two 2-cables about the knot.

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What this paper is about

It is shown that the knot invariant arising from an irreducible representation of a quantum group is, under certain conditions, an evaluation of the Homfly or Dubrovnik polynomial of the knot. Besides the known cases of the fundamental representation for each of the quantum groups in the series An, Bn, Cn and Dn, the results cover the special cases of the 3-dimensional representation of SU(2) and the 6-dimensional representation of SU(4), which can be viewed as the fundamental representations of SO(3) and SO(6) respectively. The second of these cases leads to a new relation between an evaluation of the Dubrovnik polynomial of a knot and an evaluation of the Homfly polynomials of two 2-cables about the knot.

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OpenAlex reports 7 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

It is shown that the knot invariant arising from an irreducible representation of a quantum group is, under certain conditions, an evaluation of the Homfly or Dubrovnik polynomial of the knot. Besides the known cases of the fundamental representation for each of the quantum groups in the series An, Bn, Cn and Dn, the results cover the special cases of the 3-dimensional representation of SU(2) and the 6-dimensional representation of SU(4), which can be viewed as the fundamental representations of SO(3) and SO(6) respectively. The second of these cases leads to a new relation between an evaluation of the Dubrovnik polynomial of a knot and an evaluation of the Homfly polynomials of two 2-cables about the knot.

Key concepts: Knot polynomial, HOMFLY polynomial, Quantum invariant, Mathematics, Knot invariant, Knot (papermaking), Alexander polynomial, Trefoil knot

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