Branched covers bounding rational homology balls

Abstract

Prime power fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots. In this paper, we give a new construction of non-slice knots that have the above property. The sliceness obstruction comes from computing twisted Alexander polynomials, and we introduce new techniques to simplify their calculation.

On slice alternating 3-braid closures

“…In Section 3, we consider the set S 2c \ (S 2a ∪ S 2b ∪ S 2d ∪ S 2e ) that includes strings associated to known non-χ -slice alternating 3-braid closures, such as certain Turk's head knots, and list more examples of potentially non-χ-slice knots and links. In Section 4 we follow [11] and [1] in applying a twisted Alexander polynomial obstruction to show that among these examples, three knots are indeed not slice; this concludes the classification of smoothly slice knots which are closures of alternating 3-braids with up to 20 crossings.…”

“…and let K i = βi for i = 1, 2, 3. We will show that the knots K i are not slice by adapting the approach of Aceto et al [1], based in turn on work of Herald, Kirk and Livingston [11], and demonstrating that certain reduced twisted Alexander polynomials do not factor as norms; this is a generalisation of the Fox-Milnor condition on Alexander polynomials of K i which is passed by these knots. Fix distinct primes p and q, and let ζ q denote a primitive q-th root of unity.…”

“…) coincident with two segments of L, removing those segments, joining corresponding ends along ϕ(D 1 × ∂ D 1 ) and smoothing the corners. This operation amounts to removing a 1-handle in the putative ribbon surface F. If after n band moves, the resulting link is isotopic to the (n +1)-component unlink, one has indeed obtained a ribbon surface F of Euler characteristic one bounded by L, since each component of the unlink bounds a 0-handle of F. Each band may be represented on a link diagram by an arc with endpoints on L that crosses the strands of L transversally, has no self-crossings, and is annotated by the number of half-twists in the band relative to the blackboard framing.…”

“…The remaining S 2c family is of special interest because it contains strings associated to known examples of nonslice, nonzero determinant alternating 3-braid closures, specifically Turk's head knots K 7 [19], K 11 , K 17 and K 23 [1]; the associated string of K i for i ∈ {7, 11, 17, 23} is (3 [i] ). Thus, we should not expect to find a set of band moves for all links with strings in S 2c .…”

Branched covers and rational homology balls

“…In the paper [1], Aceto, Meier, A. Miller, M. Miller, Park, and Stipsicz proved that ker ϕ contains a subgroup isomorphic to (Z 2 ) 5 . Here we will prove that ker ϕ contains a subgroup isomorphic to (Z 2 ) ∞ .…”