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arXiv · math/0311458

The algebra of knotted trivalent graphs and Turaev's shadow world

Dylan P. Thurston
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Abstract

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.

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Dylan P. Thurston. 2004-02-23. The algebra of knotted trivalent graphs and Turaev's shadow world. https://arxiv.org/abs/math/0311458

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