Team, Visitors, External Collaborators
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New Results
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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(N3/2 cw poly (n)) 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 cw , and whose components are coloured with g-modules of dimension at most N. For example, this includes the Nth-coloured Jones polynomial and the Nth-coloured HOMFLYPT polynomial.