Narrow Results

We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.

Boyer, Gordon and Watson have conjectured that an irreducible rational homology $3$-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in $S^3$ can produce large families of L-spaces, it is natural to examine the conjecture on these $3$-manifolds. Greene, Lewallen and Vafaee have proved that all $1$-bridge braids are L-space knots. In this paper, we consider three families of $1$-bridge braids. First we calculate the knot groups and peripheral subgroups. We then verify the conjecture on the three cases by applying the criterion developed by Christianson, Goluboff, Hamann and Varadaraj, when they verified the same conjecture for certain twisted torus knots and generalized the criteria due to Clay and Watson and due to Ichihara and Temma.

We show that Dehn filling on the manifold $v2503$ results in a non-orderable space for all rational slopes in the interval $(-\infty ,-1)$. This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.

In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope $r$ such that $r$-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots.

Let $M$ be a $\mathbb{Q}$-homology solid torus. In this paper, we give a cohomological criterion for the existence of an interval of left orderable Dehn surgeries on $M$. We apply this criterion to prove that the two-bridge knot that corresponds to the continued fraction $[1,1,2,2,2j]$ for $j\ge 1$ admits an interval of left orderable Dehn surgeries. This family of two-bridge knots gives some positive evidence for a question of Xinghua Gao.