Thurston, Dylan P. The algebra of knotted trivalent graphs and Turaev’s shadow world. (English) Zbl 1163.57301 Ohtsuki, T. (ed.) et al., Invariants of knots and 3-manifolds. Proceedings of the workshop, Kyoto, Japan, September 17–21, 2001. Coventry: Geometry and Topology Publications. Geom. Topol. Monogr. 4, 337-362 (2002). Summary: 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.For the entire collection see [Zbl 0996.00037]. Cited in 2 ReviewsCited in 13 Documents MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) 57M15 Relations of low-dimensional topology with graph theory 57M20 Two-dimensional complexes (manifolds) (MSC2010) 57Q40 Regular neighborhoods in PL-topology Keywords:knotted trivalent graphs; shadow surfaces; spines; simple 2-polyhedra; graph operations PDFBibTeX XMLCite \textit{D. P. Thurston}, Geom. Topol. Monogr. 4, 337--362 (2002; Zbl 1163.57301) Full Text: arXiv