×

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].

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
PDFBibTeX XMLCite
Full Text: arXiv