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Computations of quandle 2-cocycle knot invariants without explicit 2-cocycles
W Edwin Clark1, Larry A Dunning1, Masahico Saito1
1Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA.
Abstract:
We explore a knot invariant derived from colorings of corresponding 1-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle 2-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding 2-cocycles. This permits the construction of many 2-cocycle invariants without exhibiting explicit 2-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann's knot coloring polynomial. Computations using this technique show that the 2-cocycle invariant distinguishes all of the oriented prime knots up to 11 crossings and most oriented prime knots with 12 crossings including classification by symmetry: mirror images, reversals, and reversed mirrors.
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