Geometry & Topology Monographs, Vol. 4 (2002),
Invariants of knots and 3-manifolds (Kyoto 2001),
Paper no. 22, pages 337--362.

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

Dylan P. Thurston

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.

Keywords. Knotted trivalent graphs, shadow surfaces, spines, simple 2-polyhedra, graph operations

AMS subject classification. Primary: 57M25. Secondary: 57M20, 57Q40.

E-print: arXiv:math.GT/0311458

Submitted to GT on 25 November 2003. (Revised 24 January 2004.) Paper accepted 2 February 2004. Paper published 3 February 2004.

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Dylan P. Thurston
Department of Mathematics, Harvard University
Cambridge, MA 02138, USA

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