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

TL;DR

This paper explores the algebraic structure of knotted trivalent graphs (KTGs), their relation to Turaev's shadow surfaces, and how these concepts provide a new perspective on knot representations and 3-manifold topology.

Contribution

It introduces a new algebraic framework for KTGs, connects them with Turaev's shadow surfaces, and develops a Morse-theoretic approach to analyze their equivalences.

Findings

KTGs are generated by unknotted tetrahedra and Möbius strips.

Sequences of KTG operations leading to the same surface are topologically equivalent.

The approach provides a new perspective on knot representations and 3-manifold topology.

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.