An invariant of regular isotopy

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 {L_K} for a link K K , and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2. L [ unk ] = 1 {L_{[{\text {unk}}]}} = 1 . 3. L [ unk ] = a L , L [ unk ] = a − 1 L {L_{[{\text {unk}}]}} = aL,\qquad {L_{[{\text {unk}}]}} = {a^{ - 1}}L . 4. L [ unk ] + L [ unk] = z ( L [ unk] + L [ unk] ) {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 L by writhe-normalization.