[Paper Review] Framed instanton homology of surgeries on L-space knots

Tye Lidman, Juanita Pinzón‐Caicedo+1 more|arXiv (Cornell University)|Mar 6, 2020

Geometric and Algebraic Topology4 citations

TL;DR

This paper computes the framed instanton Floer homology for all integral surgeries on instanton L-space knots, proving that it matches the Heegaard Floer homology in absolute ℤ/2-grading. Using surgery exact triangles and contact topological tools, the authors establish a complete formula for $I^\#(S^3_n(K))$ in terms of the knot genus $g$, and show the Baldwin-Sivek contact invariant is homogeneous under ℤ/2-grading but not ℤ/4-grading, resolving a key structural question in instanton Floer theory for L-space surgeries.

ABSTRACT

An important class of three-manifolds are L-spaces, which are rational homology spheres with the smallest possible Floer homology. For knots with an instanton L-space surgery, we compute the framed instanton Floer homology of all integral surgeries. As a consequence, if a knot has a Heegaard Floer and instanton Floer L-space surgery, then the theories agree for all integral surgeries. In order to prove the main result, we prove that the Baldwin-Sivek contact invariant in framed instanton Floer homology is homogeneous with respect to the absolute $\mathbb{Z}/2$-grading, but not the $\mathbb{Z}/4$-grading.

Motivation & Objective

• To compute the framed instanton Floer homology $I^\#(S^3_n(K))$ for all integral surgeries on non-trivial instanton L-space knots.
• To establish a complete formula for $I^\#(S^3_n(K))$ as a ℤ/2-graded ℂ-vector space, matching the structure of Heegaard Floer homology.
• To prove that the Baldwin-Sivek contact invariant in framed instanton homology is homogeneous under ℤ/2-grading but not under ℤ/4-grading.
• To refine the computation to ℤ/4-graded groups under additional lens space surgery assumptions.
• To recover known instanton homology computations for Seifert spaces like $\Sigma(2,3,7)$ via new methods.

Proposed method

• Use of the instanton surgery exact triangle to relate $I^\#(S^3_n(K))$, $I^\#(S^3_{n+1}(K))$, and $I^\#(S^3)$, tracking ℤ/4-grading shifts via cobordism maps.
• Leverage the fact that instanton L-space knots are fibered and strongly quasipositive, with $\overline{sl}(K) = 2g(K) - 1$, to control contact structures and surgery cobordisms.
• Apply the Baldwin-Sivek contact invariant construction in framed instanton homology and analyze its grading behavior via cobordism maps.
• Use the known computation of $I^\#(S^3) \cong \mathbb{C}$ in degree 0 and the surgery triangle to inductively determine the homology groups.
• For lens space surgeries, refine the ℤ/2-grading result to ℤ/4-grading using the specific grading shifts in the surgery triangle depending on $n$.
• Verify consistency with prior results, such as $I^\#(\Sigma(2,3,7))$ and $I^\#(\Sigma(2,3,5))$, computed via different methods.

Experimental results

Research questions

• RQ1 Does the framed instanton Floer homology of integral surgeries on instanton L-space knots agree with Heegaard Floer homology in ℤ/2-grading?
• RQ2 Is the Baldwin-Sivek contact invariant in framed instanton homology homogeneous under the ℤ/4-grading?
• RQ3 Can the ℤ/4-grading of $I^\#(S^3_n(K))$ be fully computed when $S^3_n(K)$ is a lens space?
• RQ4 Do the instanton and Heegaard Floer L-space conditions yield isomorphic homology groups for all integral surgeries on knots satisfying both?
• RQ5 Is every instanton L-space $Y$ with $\dim I^\#(Y) = |H_1(Y;\mathbb{Z})|$ necessarily isomorphic to $(\lceil(n+1)/2\rceil, 0, \lfloor(n-1)/2\rfloor, 0)$ as a ℤ/4-graded vector space?

Key findings

• For any non-trivial instanton L-space knot $K$ of genus $g$, the framed instanton Floer homology $I^\#(S^3_n(K))$ is isomorphic as a ℤ/2-graded ℂ-vector space to $\widehat{HF}(S^3_n(K))$, with the explicit formula given in Theorem 1.1.
• The contact invariant $\Theta^\#$ in framed instanton homology is homogeneous under ℤ/2-grading but not under ℤ/4-grading, as shown by contradiction using cobordism maps and distinct grading shifts.
• When $S^3_n(K)$ is a lens space for $n > 0$, the ℤ/4-graded structure of $I^\#(S^3_n(K))$ is fully determined by Corollary 5.3, with explicit dimensions in each degree depending on $n$ and $g$.
• The formula for $I^\#(S^3_n(K))$ in ℤ/4-grading recovers known results for $\Sigma(2,3,7)$ and $\Sigma(2,3,5)$, confirming consistency with prior independent computations.
• For $1/n$-surgery on the right-handed trefoil with $n > 0$, the ℤ/2-graded instanton homology is $I^\#(S^3_{1/n}(T_{2,3})) \cong \mathbb{C}^n_{(0)} \oplus \mathbb{C}^{n-1}_{(1)}$, matching $\widehat{HF}$.
• The paper establishes that the isomorphism type of $I^\#(S^3_n(K))$ for instanton and Heegaard Floer L-space knots is completely determined by the genus $g$ and surgery coefficient $n$, with no torsion and minimal rank.