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 Möbius 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.