The algebra of knotted trivalent graphs and Turaev's shadow world
| dc.creator | Thurston, Dylan P. | |
| dc.date | 2003-11-25 | |
| dc.date | 2004-02-23 | |
| dc.date.accessioned | 2026-07-07T05:03:16Z | |
| dc.date.available | 2026-07-07T05:03:16Z | |
| dc.description | 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. | |
| dc.description | Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon4/paper22.abs.html | |
| dc.identifier | https://arxiv.org/abs/math/0311458 | |
| dc.identifier | http://arxiv.org/abs/math/0311458 | |
| dc.identifier | Geom. Topol. Monogr. 4 (2002) 337-362 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69349 | |
| dc.subject | Geometric Topology | |
| dc.subject | Quantum Algebra | |
| dc.subject | 57M25, 57M20, 57Q40 | |
| dc.title | The algebra of knotted trivalent graphs and Turaev's shadow world | |
| dc.type | text |