L-space surgery and twisting operation

Abstract

A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient condition for { K_n } to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family { K_n } contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one.Comment: The final version, accepted for publication by Algebr. Geom. Topo

Twist families of $L$-space knots, their genera, and Seifert surgeries

“…While no example in [45] appears to give a positive answer to this question, we still expect such a twist family to exist.…”

“…In this section we investigate Conjectures 1.2 and 1.3 from a viewpoint of braids. We have observed that for many of the twist families containing infinitely many L-space knots that are studied in [45], the twisting circle is not only a seiferter but also a braid axis. Furthermore, L-space knots are often isotopic to closures of positive or negative braids.…”

“…Twist families of Seifert fibered L-space surgeries. Common examples of twist families of surgeries containing infinitely many L-space surgeries have infinitely many L-space surgeries in which the resulting manifolds are Seifert fibered; see [45] for such examples.…”

Hyperbolic L-space knots and exceptional Dehn surgeries

“…To obtain an explicit picture of the covering knot K of κ, we apply isotopies given in Figs. [10][11][12][13][14][15]. Then taking the…”

“…Note that P (−2, 3, 2n + 1) with n ≥ 0 admits a Seifert fibered L-space surgery [10,16]. One can find a large number of twist families of hyperbolic L-space knots each of which admits a Seifert fibered L-space surgery; see [14]. To the best of our knowledge, there are no explicitly known examples of hyperbolic L-space knots which have no Seifert fibered L-space surgeries, though we expect there should be many.…”