Quandle cohomology and state-sum invariants of knotted curves and surfaces
Quandle cohomology and state-sum invariants of knotted curves and surfaces
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DOI：
10.1090/s0002-9947-03-03046-0
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发表时间：
1999-03
影响因子：
1.3
通讯作者：
J. Carter;D. Jelsovsky;D. Jelsovsky;S. Kamada;S. Kamada;Laurel Langford;M. Saito
中科院分区：
文献类型：
--
作者：
J. Carter;D. Jelsovsky;D. Jelsovsky;S. Kamada;S. Kamada;Laurel Langford;M. Saito
双扭旋转三叶结是在四维空间中打结的球面的一个例子。本文证明了这个球面与它的定向反转后的同一个球面是不同的。我们的证明基于通过racks和quandles（也称为分配广群）的上同调理论发展而来的打结曲面的态和不变量。一个quandle是一个具有二元运算的集合——其公理模拟了经典纽结理论中的赖德迈斯特移动。用quandle元素对打结曲线和曲面的图进行着色，连同quandles的上循环，被用于定义三维空间中打结圆和四维空间中打结曲面的态和不变量。本文计算了各种quandles的上同调群，并将其应用于态和不变量的研究。对于各种纽结和链环，证明了不变量的非平凡性，反之，纽结不变量被用于证明各种quandles的上同调的非平凡性。
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation - the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.