[Paper Review] L-spaces, left-orderability and two-bridge knots

Idrissa Bâ|arXiv (Cornell University)|Jan 8, 2018

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper proves that the 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ is an L-space with non-left-orderable fundamental group, verifying the L-space conjecture for this family. It further shows the 5-fold cyclic branched cover of genus one two-bridge knots $K_{[2k,-2l]}$ (for $k \geq 2$, $l > 0$) also has non-left-orderable fundamental group, completing the verification of the conjecture for all genus one two-bridge knots via the 5-fold cover.

ABSTRACT

We show that the 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ is an L-space and its fundamental group is not left-orderable. Therefore the family of 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ verifies the $L$-space conjecture. We also show that if $K_{[2k,-2l]}$ is a 2-bridge knot with $k\geq 2$, $l>0$, then the fundamental group of the 5-fold cyclic branched cover of $K_{[2k,-2l]}$ is not left-orderable, which will complete the proof that the fundamental group of the 5-fold cyclic branched cover of any genus one two-bridge knot is not left-orderable.

Motivation & Objective

• To verify the L-space conjecture for 3-fold cyclic branched covers of genus 2 two-bridge knots.
• To establish that the fundamental group of the 5-fold cyclic branched cover of genus one two-bridge knots is not left-orderable.
• To complete the proof that all genus one two-bridge knots satisfy the L-space conjecture via their 5-fold branched covers.
• To extend previous results on left-orderability and L-space properties to a broader family of two-bridge knots.

Proposed method

• Constructs the $n$-fold cyclic branched cover of a knot via gluing $n$ copies of the knot exterior along Seifert surface boundaries.
• Uses the fundamental group of the $n$-fold cyclic branched cover as the kernel of the map $\pi_1(M_K) \to \mathbb{Z}_n$.
• Applies results from Heegaard Floer homology, particularly the rank equality $\text{rk}\widehat{HF}(M) = \text{ord}(H_1(M,\mathbb{Z}))$, to identify L-spaces.
• Employs induction and recursive relations on continued fraction parameters to prove L-space properties for links $L(l;\ast,\ast,\ast)$.
• Leverages symmetry and duality in link parameters to reduce cases, using known results on quasi-alternating and alternating links.
• Applies Lemma 5.12 and Lemma 5.11 from [OSz] to propagate L-space properties through link families.

Experimental results

Research questions

• RQ1 Is the 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ an L-space with non-left-orderable fundamental group?
• RQ2 Does the 5-fold cyclic branched cover of genus one two-bridge knots $K_{[2k,-2l]}$ (for $k \geq 2$, $l > 0$) have a non-left-orderable fundamental group?
• RQ3 Can the L-space conjecture be fully verified for all genus one two-bridge knots via their 5-fold branched covers?
• RQ4 Do the 2-fold branched covers of certain quasi-alternating links have non-left-orderable fundamental groups?

Key findings

• The 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ is an L-space with non-left-orderable fundamental group, confirming the L-space conjecture for this family.
• The fundamental group of the 5-fold cyclic branched cover of $K_{[2k,-2l]}$ (for $k \geq 2$, $l > 0$) is not left-orderable, completing the proof for genus one two-bridge knots.
• The 5-fold cyclic branched cover of any genus one two-bridge knot is a total L-space, as confirmed by combining Theorem 1.3 with prior results from [DPT] and [Te].
• The 2-fold branched cover of the link $L(l;\ast,\ast,\ast)$ is an L-space for all $l > 0$, $q,s,t > 0$, establishing an infinite family of quasi-alternating links with non-left-orderable fundamental groups.
• The proof relies on inductive propagation of L-space properties through link families using symmetry and known results from Heegaard Floer homology.
• The result shows that the 3-fold branched cover of $5_1$ (a genus 2 two-bridge knot) is the Poincaré homology sphere, a known total L-space, supporting the broader conjecture.