Mathematical Research Letters

Volume 18 (2011)

Number 6

On cabled knots, Dehn surgery, and left-orderable fundamental groups

Pages: 1085 – 1095



Adam Clay (Université du Québec à Montréal, Case postale 8888, Succursale centre-ville, Montréal QC, H3C 3P8, Canada)

Liam Watson (and Department of Mathematics, University of California at Los Angeles, 520 Portola Plaza, Los Angeles CA, 90095, U.S.A.)


Previous work of the authors establishes a criterion on the fundamental group of a knot complement that determines when Dehn surgery on the knot will have a fundamental group that is not left-orderable. We provide a refinement of this criterion by introducing the notion of a $decayed$ knot; it is shown that Dehn surgery on decayed knots produces surgery manifolds that have non-left-orderable fundamental group for all sufficiently positive surgeries. As an application, we prove that sufficiently positive cables of decayed knots are always decayed knots. These results mirror properties of $L$-space surgeries in the context of Heegaard Floer homology.

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