Braided Compact Closed Categories with Applications to Low Dimensional Topology
Braided Compact Closed Categories with Applications to Low Dimensional Topology
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DOI：
10.1016/0001-8708(89)90018-2
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发表时间：
1989-10
影响因子：
1.7
通讯作者：
P. Freyd;D. Yetter
中科院分区：
文献类型：
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作者：
P. Freyd;D. Yetter
Jones [4]发现了经典链环的一个新的合痕不变量，及其随后的推广[3，7a，7 b，7 c]，重新引起了人们对打结和链环的代数表示的兴趣。受此启发，Yetter [16]分析了“缠结范畴”的结构，给出了广义缠结组成的组合描述。在这项工作中，通过对所考虑的类别使用Monoidal结构实现了极大的简化。正如我们将看到的，在[161]中的假设，即链接（或更一般地说是“马尔可夫类”）是由外在结构（即马尔可夫移动）的强加而产生的，当完整的范畴结构（包括对非对称情况的紧闭性的正确推广）被理解时，这是不必要的，这是Freyd首先提出的观察[a]。这一观察得出了新链接不变量的清晰的表示论观点，并且当与通过修改的图形“isotopy”而不是[161]中使用的标签对框架进行编码相结合时，得到了“代数化”的Kirby演算，实际上可以从手术表现中计算3-流形的不变量。
The discovery by Jones [4] of a new isotopy invariant of classical links, and its subsequent generalizations [3, 7a, 7b, 7c], has lead to renewed interest in algebraic representations of knotting and linking. Prompted by this, Yetter [16] analysed the structure of “categories of tangles” to give a combinatorial description of composition of generalized tangles. In that work a great simplification was achieved by using a monoidal structure on the categories considered. As we shall see, the assumption in [161 that links (or more generally “Markov classes”) arise by the imposition of extrinsic structure (ie, Markov moves) is unnecessary when the full categorical structure (including the correct generalization of compact closedness to the non-symmetric case) is understood, an observation first made by Freyd [a]. This observation leads to a clear representation-theoretic view of the new link invariants and, when coupled with the encoding of framings by a modified diagrammatic “isotopy” rather than the labels used in [161, to an “algebraicized” Kirby calculus from which it is actually possible to calculate invariants of 3-manifolds from their surgery presentation.