Rasmussen Spectral Sequences

• Rasmussen spectral sequences are homological constructions that connect different knot invariants through graded chain complexes and cubes-of-resolutions.
• They use sophisticated filtrations and differential structures to bridge invariants such as Khovanov, Khovanov-Rozansky, and knot Floer homologies.
• They offer practical tools for detecting properties like rank inequalities, genus, and concordance invariants in low-dimensional topology.

Rasmussen spectral sequences are a collection of homological constructions linking distinct knot invariants via graded chain complexes and filtrations. They arise in the context of categorifications of polynomial link invariants, such as Khovanov, Khovanov-Rozansky, and knot Floer homologies. Two principal spectral sequences bear the Rasmussen name: (1) the spectral sequence from reduced Khovanov-Rozansky homology to knot Floer homology as formalized in the Dunfield-Gukov-Rasmussen conjecture (Beliakova et al., 2022), and (2) the earlier spectral sequence relating Khovanov homology to knot Floer homology (Dowlin, 2018). Additional higher rank generalizations connect Homflypt, $\mathfrak{sl}_N$, and concordance invariants (Lewark, 2013). Key features include sophisticated cube-of-resolutions models, carefully defined gradings, and the collapse patterns of the associated pages ($E_r$) under filtration-preserving differentials.

1. Construction of the Homological Complexes and Cubes of Resolutions

Rasmussen spectral sequences are derived from filtered complexes associated to diagrams of knots or links. For the spectral sequence from reduced Khovanov homology to knot Floer homology (Dowlin, 2018), one constructs an oriented cube-of-resolutions complex $C^{-}_2(D)$ for a decorated braid diagram $D$:

• At each crossing, two local resolutions (oriented smoothing $D_s$ and singularization $D_x$) are constructed following specified orientation conventions.
• For each vertex $I: c(D)\rightarrow \{0,1\}$ of the cube (where $c(D)$ is the set of crossings), complete resolutions $D_I$ are defined.
• Chain groups are specified as $$C_{2}(D_I) = [R/(N(D_I)+L_I(D_I))] \otimes \mathcal L_D^+$$, where $R$ is a polynomial ring, $N(D_I)$ encodes nonlocal relations, $L_I(D_I)$ arises from local ideals, and $\mathcal L_D^+$ is a matrix factorization over $R$.
• The total differential decomposes as $d = d_0 + d_1$; $d_0$ is internal to each matrix factorization, and $d_1$ encodes edge maps in the cube, twisted by suitable sign assignments.

The gradings (homological, quantum, Hochschild, etc.) are constructed compatibly, with the cube filtration arising from the Hamming weight $|I|$ of the vertex.

2. Filtration, Differential Structure, and Identification of $E_r$ Pages

Rasmussen spectral sequences derive their structure from filtrations defined on the chain complexes:

• The filtration by cube height for the complex $C^{-}_2(D)$ ensures $d_0$ preserves filtration, while $d_1$ raises it by one.
• The associated spectral sequence satisfies $E_1^{*,*} = H_*(C^{-}_2(D), d_0)$ and converges to $H_*(C^{-}_2(D), d)$.
• The $E_1$ page can be identified as a form of reduced Khovanov homology $\overline{Kh}(K)$; edge maps in the cube correspond to Khovanov Frobenius maps up to scaling.
• The $E_\infty$ page corresponds to the hat-version of knot Floer homology $\widehat{HFK}(K)$ after reduction and setting variables appropriately.

Differentials $d_r$ beyond $d_1$ are constrained by parity grading: even-indexed differentials vanish, and odd-indexed differentials shift gradings compatibly.

3. Rasmussen’s Spectral Sequence: Khovanov to Knot Floer Homology

The original Rasmussen spectral sequence verifies a conjectured rank inequaility:

$$\mathrm{rank}_Q\ \overline{Kh}(K)\ \ge\ \mathrm{rank}_Q\ \widehat{HFK}(K)$$

for all knots $K\subset S^3$ (Dowlin, 2018). The spectral sequence begins at $E_2 = \overline{Kh}(K)$ and abuts at $E_\infty = \widehat{HFK}(K)$. The construction and convergence are guaranteed by the compatibility of cube filtration and chain maps. Examples include the trefoil and unknot, where explicit calculation of differentials confirms the structure and rank inequality.

4. The DGR Spectral Sequence and gl$(0)$-Knot Homology

The proof of the Dunfield-Gukov-Rasmussen conjecture (Beliakova et al., 2022) involves two interlocking spectral sequences:

• From the reduced triply-graded Khovanov-Rozansky homology $HHH^{p,q,*}_{red}(K)$, introduce a differential $d_0$ of tri-degree $(-2, 0, 1)$ on each vertex-space, anticommute with the cube differential, and obtain

$$E_1^{p,q} \simeq HHH^{p,q,*}_{red}(K),\qquad E_2^{p,q} \simeq H^{p,q}_{\mathfrak{gl}_0}(K)$$

• From $H^{p,q}_{\mathfrak{gl}_0}(K)$ (the $\mathfrak{gl}_0$ homology defined via cube-of-resolutions and quotient constructions), define a Bockstein spectral sequence by manipulating coefficients (e.g., specializing $q=1$ and considering nonlocal trace relations). The differentials $\delta_r$ shift quantum grading by $+1$ per page.

The composite spectral sequence yields: $$E_1^{p,q} = HHH^{p,q,*}_{red}(K) \Longrightarrow E_\infty^{p,q} \simeq \widehat{HFK}^{p+q,*}(K)$$ All differentials, gradings, and module structures are compatible with homological, quantum, and Alexander conventions.

5. Higher Rank, Homflypt, and sl$(N)$ Connections

Generalizations of Rasmussen spectral sequences relate various categorified polynomial invariants:

• A spectral sequence with differentials $d_k^R$ of tri-degree $(1, 2Nk, -2k)$ starting from triply-graded reduced Homflypt homology converges to regraded reduced $\mathfrak{sl}_N$ homology (Lewark, 2013).
• Lee-Gornik’s deformation provides a filtration on Khovanov-Rozansky complexes with differentials of bidegree $(1,1)$, yielding concordance invariants $s_N(K)$ for knots.
• An additional spectral sequence relates reduced and unreduced $\mathfrak{sl}_N$ homology via an increasing filtration induced by powers of $X$ in $\mathbb{C}[X]/(X^N)$.

For each fixed $N$, the collection of concordance invariants $\{s_N\}_{N\ge2}$ defines a linearly independent family of homomorphisms on the knot concordance group. Concrete calculations, e.g., for pretzel knots, confirm independence and sharpness of bounds extracted from the sequence structure.

6. Knot Detection and Applications

Using the collapse patterns and grading supports, certain knots are detected by the spectral sequence structure:

• The unknot, trefoils, figure-eight, and cinquefoil are detected via the support of reduced triply-graded Khovanov-Rozansky and $\mathfrak{gl}_0$ homology as corollaries of spectral sequence structure (Beliakova et al., 2022).
• The rank inequality supports fiberedness and genus detection via knot Floer homology (Dowlin, 2018).
• Slice-genus bounds and combinatorial inequalities are extracted from the algebraic relations induced by decategorification at each page, as in (Lewark, 2013).

7. Convergence, Differentials, and Homological Algebra

Convergence of Rasmussen spectral sequences is established through filtered quasi-isomorphism and homological algebra:

• If a filtered chain map induces isomorphism on associated graded homologies, all higher pages agree.
• The cube-filtration and parity grading ensure only odd differentials occur, and for small knots (with homology support in a single bigrading), all differentials vanish, so the spectral sequences collapse early.
• All higher Bockstein differentials in gl$(0)$ sequences have the same quantum shift and collapse after finitely many pages by PID arguments.

These mechanisms justify the use of the spectral sequences for detecting topological knot invariants and relating different homological approaches within low-dimensional topology.