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

Dylan P Thurston

Geometry & Topology Monographs 4 (2002) 337–362

DOI: 10.2140/gtm.2002.4.337

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 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.

Keywords

knotted trivalent graphs, shadow surfaces, spines, simple 2-polyhedra, graph operations

Mathematical Subject Classification

Primary: 57M25

Secondary: 57M20, 57Q40

References
Publication

Received: 25 November 2003
Revised: 24 January 2004
Accepted: 2 February 2004
Published: 3 February 2004

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