Clark W Edwin, Dunning Larry A, Saito Masahico
Department of Mathematics and Statistics, University of South Florida, Tampa, FL, USA.
J Knot Theory Ramif. 2017 Jun;26(7). doi: 10.1142/s0218216517500353. Epub 2017 Mar 22.
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.
我们探索一种从具有任意连通 quandles 的相应 1 - 缠结的着色导出的纽结不变量。当 quandle 是某类阿贝尔扩张时，该不变量等同于 quandle 2 - 上循环不变量。我们使用广义亚历山大 quandles 构造了许多这样的阿贝尔扩张，而无需明确找到 2 - 上循环。这使得我们能够构造许多 2 - 上循环不变量，而无需展示明确的 2 - 上循环。我们表明，对于连通的广义亚历山大 quandles，该不变量等同于艾泽曼的纽结着色多项式。使用此技术的计算表明，2 - 上循环不变量可以区分所有交叉数达 11 的有向素纽结以及大多数交叉数为 12 的有向素纽结，包括通过对称性进行分类：镜像、反转和反转镜像。
