Bounds on alternating surgery slopes

Abstract

We show that if p/q-surgery on a nontrivial knot K yields the branched double cover of an alternating knot, then |p/q| ≤ 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 S 3 p/q (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 p/q and the Betti number b 2 (X).

The realization problem for noninteger Seifert fibered surgeries

“…Remark 4.7 It is clear from (4-2) that the vector w 0 determines the V i . It turns out that the sequence of V i , along with (4-2), is sufficient to determine the stable coefficients of w 0 [21]. In particular, this means that the intersection form Q X is determined by the knot, the surgery slope p=q and the second Betti number of X .…”

The realization problem for non-integer Seifert fibered surgeries

“…It is clear from (5.2) that the vector w 0 determines the V i . It turns out that the sequence of V i along with equation (5.2) is sufficient to determine the stable coefficients of w 0 [McC17]. In particular, this means that the intersection form Q X is determined by the knot, the surgery slope p/q and the second Betti number of X.…”