Non-braid-positive hyperbolic L-spaceknots

An L-space knot is a knot that admits a positive Dehn surgery yielding an L-space.Many known hyperbolic L-space knots are braid positive, meaning they can be represented as the closure of a positive braid.Recently, Baker and Kegel showed that the hyperbolic L-space knot o9_30634 from Dunfield's census is not braid-positive, and they constructed infinitely many candidates for hyperbolic L-space knots that may not be braid-positive.However, it remains unproven whether their examples are genuinely non-braid-positive.In this paper, we construct infinitely many hyperbolic L-space knots that are not braid-positive, and are distinct from those considered by Baker and Kegel. IntroductionAn L-space is a rational homology 3-sphere whose (hat version) Heegaard Floer homology has rank equal to the order of its first homology.A knot is called an L-space knot if it admits a positive Dehn surgery yielding an L-space.L-space knots were originally motivated by the study of knots admitting lens space surgeries [14], and they continue to be an active subject of research.In this paper, we consider the braid positivity of L-space knots.A knot or link is said to be braid-positive if it can be expressed as the closure of a positive braid.Many known L-space knots are braid-positive; for example, positive torus knots are L-space knots, and then they are braid-positive.On the other hand, it is known that not all L-space knots are braid-positive; for example, the (2, 3)-cable of the right-handed trefoil is not braid-positive (see Example 1 of [1]).However, the existence of non-braid-positive hyperbolic L-space knots remained an open problem for some time (see Problem 31.2 of [5], Question 2 of [1]).Recently, Baker and Kegel examined Dunfield's list of 632 hyperbolic L-space knots.They showed that all but one of these knots are braid-positive, and that the exceptional knot, o9_30634, is not braid-positive [2].Furthermore, they constructed infinitely many candidates for non-braid-positive hyperbolic L-space knots, although they could not prove that any of their examples are definitely non-braid-positive.