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Section: New Results

Algorithmic and Combinatorial Aspects of Low Dimensional Topology

Treewidth, crushing and hyperbolic volume

Participant : Clément Maria.

In collaboration with Jessica S. Purcell (Monash University, Australia)

The treewidth of a 3-manifold triangulation plays an important role in algorithmic 3-manifold theory, and so it is useful to find bounds on the tree-width in terms of other properties of the manifold. In [26], we prove that there exists a universal constant $c$ such that any closed hyperbolic 3-manifold admits a triangulation of tree-width at most the product of $c$ and the volume. The converse is not true: we show there exists a sequence of hyperbolic 3-manifolds of bounded tree-width but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects tree-width by at most a constant multiple.

Parameterized complexity of quantum knot invariants

Participant : Clément Maria.

In [53], we give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a $O\left(N^{3/2\mathrm{cw}}\mathrm{poly}\left(n\right)\right)$ time algorithm to compute any Reshetikhin-Turaev invariant-derived from a simple Lie algebra $g$ of a link presented by a planar diagram with $n$ crossings and carving-width $\mathrm{cw}$, and whose components are coloured with $g$-modules of dimension at most $N$. For example, this includes the $N$th-coloured Jones polynomial and the $N$th-coloured HOMFLYPT polynomial.