Geometric Topology

Fatgraph Algorithms and the Homology of the Kontsevich Complex

by Riccardo Murri

Pre-print available on arXiv: http://arxiv.org/abs/1202.1820

Fatgraphs are multigraphs enriched with a cyclic order of the edgesvincident to a vertex.  This paper presents... more Fatgraphs are multigraphs enriched with a cyclic order of the edgesvincident to a vertex.  This paper presents algorithms to:
(1) generate the set of fatgraphs, given the genus and the number of boundary cycles;
(2) compute automorphisms of any given fatgraph;
(3) compute the homology of the fatgraph complex.
The algorithms are suitable for effective computer implementation.

In particular, this allows us to compute the rational homology of
the moduli space of Riemann surfaces with marked points. We thus compute the Betti numbers of $M_{g,n}$ with $(2g + n) \leq 6$.

An upper bound on Reidemeister moves

by Alexander Coward

Submitted

We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the... more We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links.

Unknotting genus one knots

by Alexander Coward

Comment. Math. Helv. 86 (2011) 383-399

For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is... more For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are precisely two unknotting crossing changes. The proof uses sutured manifold theory and an analysis of the arc complex of the once-punctured torus.

Algorithmically detecting the bridge number of hyperbolic knots

by Alexander Coward

We show that, up to ambient isotopy, the exterior of a hyperbolic knot in the 3-sphere admits finitely many bridge... more We show that, up to ambient isotopy, the exterior of a hyperbolic knot in the 3-sphere admits finitely many bridge punctured 2-spheres of given Euler characteristic and that there is an algorithm to find all of these surfaces. This yields an algorithm to detect bridge number for hyperbolic knots.

Ordering the Reidemeister moves of a classical knot

by Alexander Coward

Alg. Geom. Top. 6 (2006) 659-671

We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are... more We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are sorted by type.

The curve complex and covers via hyperbolic 3-manifolds

by Robert Tang