We classify all the exceptional Dehn surgeries on the minimally twisted chain links with six and seven components. arXiv:1808.08430v2 [math.GT]

Abstract. We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of "weight systems", finding everything to be in agreement with the conjecture that "every weight system integrates".

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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 realise the maximal known distances between slopes corresponding to exceptional (lens, lens), (lens, toroidal) and (lens, Seifert fibred) pairs. In light of Baker's work, the classification in this paper conjecturally accounts for "most" hyperbolic knots in S 3 realising the maximal distance between these exceptional pairs. As a byproduct, we obtain that all examples that arise from the 5-chain link actually arise from 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.