Volume 4 (2002)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
All Volumes
About this Series
Ethics Statement
Purchase Printed Copies
Author Index
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
MSP Books and Monographs
Other MSP Publications

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


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.


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

Mathematical Subject Classification

Primary: 57M25

Secondary: 57M20, 57Q40


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

Dylan P Thurston
Department of Mathematics
Harvard University
Cambridge MA 02138