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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

An invariant of regular isotopy


Author: Louis H. Kauffman
Journal: Trans. Amer. Math. Soc. 318 (1990), 417-471
MSC: Primary 57M25
DOI: https://doi.org/10.1090/S0002-9947-1990-0958895-7
MathSciNet review: 958895
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Abstract: This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted $ {L_K}$ for a link $ K$, and it satisfies the axioms:

1. Regularly isotopic links receive the same polynomial.

2. $ {L_{[{\text{unk}}]}} = 1$.

3. $ {L_{[{\text{unk}}]}} = aL,\qquad {L_{[{\text{unk}}]}} = {a^{ - 1}}L$.

4. $ {L_{[{\text{unk}}]}} + {L_{[{\text{unk]}}}} = z({L_{[{\text{unk]}}}} + {L_{[{\text{unk]}}}})$.

Small diagrams indicate otherwise identical parts of larger diagrams.

Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from $ L$ by writhe-normalization.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0958895-7
Article copyright: © Copyright 1990 American Mathematical Society