Mathematics Lecture Notes

Theorem8.1Unknot Detection by Khovanov Homology (Kronheimer-Mrowka)

Let $K$ be a knot in $S^3$ . If the reduced Khovanov homology of $K$ has total rank 1 (i.e., $\widetilde{\mathrm{Kh}}(K) \cong \mathbb{Q}$ ), then $K$ is the unknot. Equivalently, Khovanov homology detects the unknot: $K$ is unknotted if and only if $\widetilde{\mathrm{Kh}}(K) \cong \widetilde{\mathrm{Kh}}(U)$ .

Proof

The proof by Kronheimer and Mrowka (2011) uses gauge theory (specifically, a variant of instanton Floer homology) and proceeds through several deep results.

Step 1: Singular instanton homology. Kronheimer and Mrowka define singular instanton knot homology $I^\natural(K)$ , a gauge-theoretic invariant associated to a knot $K \subset S^3$ . This is constructed using the Chern-Simons functional on connections with prescribed singular behavior along $K$ : solutions to the anti-self-duality equations $F_A^+ = 0$ on $\mathbb{R}\times(S^3\setminus K)$ with specific holonomy conditions around $K$ .

Step 2: Spectral sequence from Khovanov to instanton homology. They construct a spectral sequence $E_2 = \widetilde{\mathrm{Kh}}(K) \Rightarrow I^\natural(K)$ . This means: $\mathrm{rank}\,\widetilde{\mathrm{Kh}}(K) \geq \mathrm{rank}\,I^\natural(K)$ . The spectral sequence is constructed by analyzing the instanton equations on cobordisms corresponding to the cube of resolutions in the Khovanov complex.

Step 3: Instanton homology detects the unknot. They prove that $I^\natural(K)$ detects the unknot: $I^\natural(K) \cong \mathbb{C}$ if and only if $K$ is the unknot. The proof uses:

The relationship between $I^\natural(K)$ and the $\mathrm{SU}(2)$ character variety of the knot group (representations $\pi_1(S^3\setminus K) \to \mathrm{SU}(2)$ ).
For the unknot: $\pi_1(S^3\setminus U) = \mathbb{Z}$ , so the character variety is essentially trivial, giving $I^\natural(U) \cong \mathbb{C}$ .
For nontrivial knots: a result of Kronheimer-Mrowka (building on work of Fintushel-Stern and Floer) shows that $I^\natural(K) \neq \mathbb{C}$ by detecting nontrivial $\mathrm{SU}(2)$ representations, which exist by the Property P theorem and its generalizations.

Step 4: Combining. If $\widetilde{\mathrm{Kh}}(K) \cong \mathbb{Q}$ (rank 1), then the spectral sequence gives $\mathrm{rank}\,I^\natural(K) \leq 1$ . Since $I^\natural(K)$ is always nontrivial ( $\mathrm{rank} \geq 1$ ), we get $\mathrm{rank}\,I^\natural(K) = 1$ , which implies $K$ is the unknot. $\square$

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ExampleKhovanov Homology of Nontrivial Knots

Trefoil: $\widetilde{\mathrm{Kh}}(3_1)$ has rank 2 (two generators in different bigradings), immediately proving $3_1$ is nontrivial. Figure-eight: $\widetilde{\mathrm{Kh}}(4_1)$ has rank 3. Note that the Alexander polynomial $\Delta_{4_1}(t) = -t+3-t^{-1}$ also detects these, but there exist nontrivial knots with trivial Alexander polynomial for which Khovanov homology is still nontrivial (e.g., the Kinoshita-Terasaka knot has $\Delta = 1$ but $\mathrm{rank}\,\widetilde{\mathrm{Kh}} > 1$ ).

RemarkRelation to Other Detection Results

Knot Floer homology $\widehat{HFK}(K)$ (Ozsvath-Szabo) also detects the unknot, the trefoils, and the figure-eight knot, and detects the Seifert genus and fibredness. The analogous spectral sequence $\widetilde{\mathrm{Kh}}(K) \Rightarrow \widehat{HF}(\Sigma(K))$ (from Khovanov homology to the Heegaard Floer homology of the branched double cover) was constructed by Ozsvath-Szabo. Whether the Jones polynomial alone detects the unknot remains one of the major open problems in knot theory. Khovanov homology's unknot detection is currently the strongest result in this direction.

Math Notes

Khovanov Homology Detects the Unknot

Proof (Outline)