#### 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.

Notes on file formats
Dylan P. Thurston

Department of Mathematics, Harvard University

Cambridge, MA 02138, USA

Email: dpt@math.harvard.edu

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