Volume 17, issue 5 (2017)

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Bounds on alternating surgery slopes

Duncan McCoy

Algebraic & Geometric Topology 17 (2017) 2603–2634

We show that if pq–surgery on a nontrivial knot K yields the branched double cover of an alternating knot, then |pq| 4g(K) + 3. This generalises a bound for lens space surgeries first established by Rasmussen. We also show that all surgery coefficients yielding the double branched covers of alternating knots must be contained in an interval of width two and this full range can be realised only if the knot is a cable knot. The work of Greene and Gibbons shows that if Spq3(K) bounds a sharp 4–manifold X, then the intersection form of X takes the form of a changemaker lattice. We extend this to show that the intersection form is determined uniquely by the knot K, the slope pq and the Betti number b2(X).

alternating knots, branched double covers, Dehn surgery
Mathematical Subject Classification 2010
Primary: 57M12, 57M25
Secondary: 57M27
Received: 15 December 2014
Revised: 27 February 2017
Accepted: 25 March 2017
Published: 19 September 2017
Duncan McCoy
Department of Mathematics
University of Texas
Austin, TX
United States