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Fionntan Roukema

Publications and source records attributed to Fionntan Roukema.

5 recordsLinked to original sources
arXiv2017

On hyperbolic knots in S^3 with exceptional surgeries at maximal distance

Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal), (lens, Seifert fibred spaces) pairs. In light of Baker's work, the classification in this paper conjecturally accounts for 'most' hyperbolic knots in S^3 realizing the maximal distance between these exceptional pairs. All examples obtained in our classification are realized by filling the magic manifold. The classification highlights additional examples not mentioned in Martelli and Petronio's survey of the exceptional fillings on the magic manifold. Of particular interest, is an example of a knot with two lens space surgeries that is not obtained by filling the Berge manifold.

math.GT↗
arXiv2015

Exceptional Slopes on Manifolds of Small Complexity

It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic $3$-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete classification of the sets of exceptional slopes and fillings for all cusped hyperbolic surgeries on the minimally twisted 5-chain link, thereby describing the sets of exceptional slopes and fillings for most hyperbolic manifolds of small complexity. The classification produces the description of exceptional fillings for many families of one and two cusped manifolds, and provides supporting evidence for some well-known conjectures. One such family that appears in the classification is an infinite family of 1-cusped hyperbolic manifolds with four Seifert manifold fillings and a toroidal filling.

math.GT↗
arXiv2013

Exceptional Dehn surgery on the minimally twisted five-chain link

We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or fewer (including 0) cusps. In view of Thurston's hyperbolic Dehn filling theorem it is then natural to face the problem of classifying all the exceptional fillings on M5, namely those yielding non-hyperbolic 3-manifolds. Here we completely solve this problem, also showing that, thanks to the symmetries of M5 and of some hyperbolic manifolds resulting from fillings of M5, the set of exceptional fillings on M5 is described by a very small amount of information.

math.GT↗
arXiv2007

Goussarov-Polyak-Viro combinatorial formulas for finite type invariants

Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result implies the existence of Gauss diagram combinatorial formulas for finite type invariants. This article presents a simplified account of the original approach. The simplifications provide an easy generalization to the cases of pure tangles and pure braids. The associated problem on group algebras is introduced and used to prove the existence of ``multi-local word formulas'' for finite type invariants of pure braids.

math.GT↗