[Paper Review] Surgery obstructions from Khovanov homology

Liam Watson|arXiv (Cornell University)|Jul 9, 2008

Geometric and Algebraic Topology44 references22 citations

TL;DR

This paper establishes new obstructions to exceptional Dehn surgeries—specifically lens space and finite fundamental group surgeries—on strongly invertible knots in $S^3$ using Khovanov homology. By analyzing the homological width of branch sets in the two-fold branched cover, it shows that Khovanov homology can detect when surgery obstructions arise, with only finitely many Khovanov homology computations often sufficient to rule out such surgeries.

ABSTRACT

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S^3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.

Motivation & Objective

• To develop new obstructions to exceptional Dehn surgeries on strongly invertible knots in $S^3$ using Khovanov homology.
• To connect the geometric complexity of 3-manifolds obtained by surgery to the homological width of their branch sets in the two-fold branched cover.
• To provide computationally feasible criteria—based on Khovanov homology—that can rule out finite or lens space fillings without requiring full knot Floer homology.
• To demonstrate that Khovanov homology can detect obstructions even when classical invariants like the Alexander polynomial fail.
• To establish a correspondence between geometric simplicity in branched covers and algebraic simplicity (thin homology) in Khovanov homology of the branch set.

Proposed method

• Use the two-fold branched cover construction to relate surgeries on a strongly invertible knot $K$ to closures of its associated quotient tangle $T$.
• Apply the result that lens spaces and manifolds with finite fundamental group arise only as branched covers of homologically thin links.
• Define homological width in Khovanov homology as a measure of complexity: links with width greater than two diagonals obstruct finite or lens space fillings.
• Leverage a stable behavior of Khovanov homology under rational tangle attachments (Lemma 4.10) to reduce the number of required computations to a finite set.
• Use the spectral sequence between Khovanov homology and Heegaard Floer homology (Ozsváth–Szabó) to justify the geometric relevance of Khovanov invariants.
• Apply the obstruction via reduced Khovanov homology: if the homology is supported in more than two adjacent diagonals, no finite or lens space filling is possible.

Experimental results

Research questions

• RQ1 Can Khovanov homology detect obstructions to lens space surgeries on strongly invertible knots in $S^3$?
• RQ2 To what extent can homological width in Khovanov homology replace or complement Heegaard Floer homology in obstructing exceptional surgeries?
• RQ3 Are there computationally efficient criteria based on Khovanov homology that can rule out finite fundamental group fillings?
• RQ4 How does the Khovanov homology of branch sets relate to the geometry of the resulting 3-manifolds after Dehn surgery?
• RQ5 Can Khovanov homology provide stronger or more accessible obstructions than the Alexander polynomial for finite fillings?

Key findings

• For a strongly invertible knot $K$, if the reduced Khovanov homology of the branch set (from rational tangle closures of the quotient tangle) is supported in more than two adjacent diagonals, then $K$ does not admit a non-trivial finite fundamental group surgery.
• The same condition—homological width greater than two—rules out lens space surgeries, provided the branch set is homologically thin in the required way.
• For the 14-crossing non-alternating knot $14^n_{11893}$, only two Khovanov homology computations ($n = -7$ and $n = -9$) were needed to confirm $w_{ ext{min}} = w_{ ext{max}} = 4$, proving it admits no finite fillings.
• The Khovanov homology of the branch set for $S^3_n(K)$ at $n = -7$ is $ ilde{ ext{Kh}} o bF^{20} igoplus bF^{36} igoplus bF^{39} igoplus bF^{16}$, with Euler characteristic 7, supporting the width argument.
• The method is effective even when the Alexander polynomial does not obstruct finite fillings, as in the case of $14^n_{11893}$, where $ riangle_K(t) = t^{-3} - t^{-2} + t^{-1} - 1 + t - t^2 + t^3$.
• The approach provides a practical alternative to full knot Floer homology, with Khovanov homology computations often sufficient to rule out finite or L-space fillings.