Brieskorn spheres bounding rational balls

Abstract

Fintushel and Stern showed that the Brieskorn sphere Σ(2, 3, 7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls: the families Σ(2, 4n + 1, 12n + 5) and Σ(3, 3n + 1, 12n + 5) for n odd. We also provide handlebody diagrams for a rational homology ball containing a rationally slice disk for the figure-eight knot, as well as for a rational homology ball bounded by Σ(2, 3, 7). These handle diagrams necessarily contain 3-handles.

Universal Quantum Computing and Three-Manifolds

“…Exceptional Dehn surgery at slope (0, 1) on the figure-of-eight knot K 0 leads to a remarkable manifold Σ Y found in [39] in the context of 3-dimensional integral homology spheres smoothly bounding integral homology balls. Apart from its topological importance, we find that some of its coverings are associated to already discovered ICs and those coverings have the same fundamental group π 1 (Σ Y ).…”

“…We will start with our friend T 1 and arrive at a few standard 3-manifolds of historic importance, the Poincaré homology sphere [alias the Brieskorn sphere Σ(2, 3, 5)], the Brieskorn sphere Σ(2, 3, 7) and a Seifert fibered toroidal manifold Σ . Then we introduce the 3-manifold Σ Y resulting from 0-surgery on the figure-of-eight knot [39]. Later in this section, we will show how to use the {3, 5, 3} Coxeter lattice and surgery to arrive at a hyperbolic 3-manifold Σ 120e of maximal symmetry whose several coverings (and related POVMs) are close to the ones of the trefoil knot [40].…”

“…The sphere Σ(2, 3, 5) may be identified to the Poincaré homology sphere. The sphere Σ(2, 3, 7) [39] may be obtained from (1, 1) surgery on T 1 . Table 5 provides the sequences η d for the corresponding surgeries (±1, 1) on T 1 .…”

A note on PL-disks and rationally slice knots

“…Remark 1.3. The figure-eight satisfies the hypotheses of Theorem 1.2 by [FS84] (see also [AL18, Section 3] and [HM17, Theorem 1.7]). More generally, the genus one knots K n with n positive full twists in one band and n negative full twists in the other band, n odd, also satisfy the hypotheses of Theorem 1.2; see Figure 1.…”

Using rational homology circles to construct rational homology balls

“…Understanding which rational homology 3-spheres (QS 3 s) bound rational homology 4-balls (QB 4 s) is a widely explored open question among Kirby's list of problems (Problem 4.5 in [1]). Certain classifications of QS 3 s bounding QB 4 s do exist (e.g., lens spaces [17], certain small Seifert fibered spaces [15], some Dehn surgeries on knots [3], and some Brieskorn spheres [4], [10], [6]), but the question at large is far from resolved. In [4], Akbulut-Larson used the fact that 0-surgery on the figure-eight knot bounds a rational homology circle (QS 1 × B 3 ) to construct infinite families of Brieskorn spheres that bound QB 4 s. Their construction relies in part on the following lemma, which Akbulut-Larson proved for the case of 0-surgery on the figure-eight knot (and, more generally, rationally slice knots).…”