三价打结图的代数与Turaev的影子世界
The algebra of knotted trivalent graphs and Turaev's shadow world
摘要：
打结三价图（KTG）形成了一个丰富的代数，只有一些简单的操作：连接和，解压缩和冒泡。通过这些操作，KTG由未打结的四面体和Moebius带生成。许多以前已知的表示结，包括结图和非关联缠结，可以变成KTG表示在一个自然的方式。通常，两个KTG操作序列在所有输入上产生相同的输出。这些“基本”关系可能是微妙的：例如，有一个平面代数的KTG与一个杰出的循环。研究这些关系自然会让我们想到图拉耶夫的阴影曲面，这是一种基于简单的4-流形的2-样条的3-流形的组合表示。我们认为打结的三价图的边界，这样一个简单的脊椎的4球，并考虑莫尔斯理论sweepout的脊椎作为一个'电影'的打结图，因为它根据KTG操作的演变。对于每一个KTG呈现的结，我们可以构建这样一个电影。产生相同曲面的两个KTG运算序列在拓扑上是等价的，尽管反过来并不完全正确。
英文摘要：
Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be turned into KTG presentations in a natural way. Often two sequences of KTG operations produce the same output on all inputs. These `elementary' relations can be subtle: for instance, there is a planar algebra of KTGs with a distinguished cycle. Studying these relations naturally leads us to Turaev's shadow surfaces, a combinatorial representation of 3-manifolds based on simple 2-spines of 4-manifolds. We consider the knotted trivalent graphs as the boundary of a such a simple spine of the 4-ball, and to consider a Morse-theoretic sweepout of the spine as a `movie' of the knotted graph as it evolves according to the KTG operations. For every KTG presentation of a knot we can construct such a movie. Two sequences of KTG operations that yield the same surface are topologically equivalent, although the converse is not quite true.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
