Collapse and Involutions in the Khovanov-Floer Spectral Sequence

Definitions and mathematical statement

This record presents a collection of questions rather than a single conjecture. Let $K$ be a knot to which the source's involutive Khovanov--Heegaard Floer spectral sequence applies, and denote that spectral sequence by $(E_r(K),d_r)$. Write $\Sigma_2(K)$ for the double cover branched along $K$, in the ambient setting intended by the source. The supplied record does not specify that ambient setting, the coefficient ring, the initial page, the gradings, or the precise version of involutive Heegaard Floer homology at the target. Consequently, no particular identification of the initial or limiting page is asserted here.

For a spectral sequence with its indexing and convergence fixed, say that it collapses at page $r$ if $d_s=0$ for every defined page $s\ge r$. Under the usual convergence hypotheses, this identifies $E_r$ with the limiting page $E_\infty$; recovery of the target may additionally require resolving filtration extensions. Interpret immediate collapse as collapse at the first page intended in the source, whose index is not supplied.

The questions are:

1. Describe a larger class of knots for which this spectral sequence collapses immediately. The supplied dated assessment states that collapse is known for quasi-alternating knots, providing a possible baseline for this question. However, the excerpt does not explicitly identify the class relative to which "larger" is meant, nor does it specify the collapse page for the stated quasi-alternating cases.
2. Determine what geometric or topological information is conveyed by the first page at which the spectral sequence collapses, including whether that page is independent of the choices in its construction.
3. Determine conditions on $K$ under which this spectral sequence yields information about the involutive Heegaard Floer homology of $\Sigma_2(K)$, and specify the information obtained. The original excerpt asks for information about this homology; the cleaned statement also asks when the homology can be determined. These are different strengths of conclusion and are retained as such.
4. Let $D$ be a knot diagram and let $(C_{\mathrm{Kh}}(D),d)$ denote the Khovanov chain complex in the conventions intended by the source. Does the object denoted $\widetilde{BN}$ in the cleaned statement arise from a homotopy involution on some chain homotopy representative of this complex? Here a homotopy involution means a chain endomorphism $\iota$ satisfying \[ d\iota=\iota d,\qquad \iota^2\simeq\mathrm{id}. \] With a homological differential, the latter condition means that there is a degree $+1$ map $H$ with \[ \iota^2-\mathrm{id}=dH+Hd. \] The source does not specify what construction is meant by "arise from," or what notion of nontriviality is intended. In particular, whether nontriviality means $\iota\not\simeq\mathrm{id}$ requires clarification. The original excerpt writes $\mathrm{BNg}$, whereas the cleaned statement writes $\widetilde{BN}$; their identification is unresolved in the supplied evidence.
5. Determine the chain endomorphism of $C_{\mathrm{Kh}}(D)$ intended to result from reflection followed by crossing reversal, and determine whether it is a nontrivial homotopy involution. This geometric description occurs in the cleaned statement, while the original excerpt asks for an induced endomorphism without specifying the operation. The reflection, crossing-reversal convention, and identifications needed to obtain an endomorphism of the original complex are not supplied.

These questions remain conditional on the missing construction conventions and notation; the supplied record does not determine a unique fully specified formulation.

Short English statement

The problem asks which knots have immediate collapse of the involutive Khovanov--Heegaard Floer spectral sequence, what the collapse page measures, and when the sequence gives information about the branched double cover's involutive Floer homology. It also asks whether the indicated $BN$ construction and a diagram operation come from nontrivial homotopy involutions of the Khovanov complex. Several conventions and the relationship between $\mathrm{BNg}$ and $\widetilde{BN}$ remain unspecified.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30005381$, source problem code catalogue entry, title "Collapse and Involutions in the Khovanov-Floer Spectral Sequence." The record attributes the questions to the Oberwolfach Reports workshop collection, with proposed year 2023; no individual proposer is supplied.
• Morphisms in Low Dimensions (2023), Oberwolfach Reports, report 4/2023, pp. 215--259, DOI 10.4171/owr/2023/4. Supplied report reference: https://doi.org/10.4171/owr/2023/4.
• Alishahi, Truong, and Zhang, Involutive branched-cover spectral sequence, as cited in the supplied literature sources: https://arxiv.org/abs/2305.07172.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open, states that the involutive Khovanov--Heegaard Floer spectral sequence is constructed and collapses for quasi-alternating knots, and describes the general collapse, page, functoriality, and chain-involution questions as remaining open. This is supplied dated curation, not an independently verified literature assessment. Its stated instruction to verify time-sensitive status and cited claims independently remains applicable.
• The registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report. That verification claim is supplied metadata; no source was independently consulted for this draft.