Positivity and Evenness in HOMFLY-PT Knot Homology

Definitions and mathematical statement

The supplied record asks for two classifications of knots. Its polynomial normalization, homology conventions, and intended category of knots are not specified, so the following formulation is conditional on fixing those conventions.

Let $\mathcal K$ denote the intended set of knot isotopy classes. For $K\in\mathcal K$, let $P_K(q,a)$ denote its HOMFLY--PT invariant in a fixed normalization. If this normalization gives an integral Laurent polynomial, write uniquely \[ P_K(q,a)=\sum_{(i,j)\in{\mathbb{Z}}^2}c_{i,j}(K)q^i(-a)^j, \qquad c_{i,j}(K)\in{\mathbb{Z}}, \] with finitely many nonzero coefficients. Interpret coefficient positivity provisionally as $c_{i,j}(K)\geq 0$ for every $i,j$, equivalently strict positivity for every nonzero coefficient. The first requested classification is to determine \[ \mathcal P=\{K\in\mathcal K: c_{i,j}(K)\geq 0\text{ for every }(i,j)\in{\mathbb{Z}}^2\}. \] This definition requires clarification if the intended invariant is rational rather than Laurent polynomial, or if positivity refers to a particular expansion.

After fixing a reduced or unreduced version of triply graded Khovanov--Rozansky homology, its coefficient ring, and its absolute grading conventions, write \[ \mathcal H(K)=\bigoplus_{i,j,k\in{\mathbb{Z}}}\mathcal H^{i,j,k}(K), \] where $k$ denotes the homological degree. The second requested classification is to determine \[ \mathcal E=\{K\in\mathcal K:\mathcal H^{i,j,k}(K)=0 \text{ for all }i,j\in{\mathbb{Z}}\text{ and all }k\in 2{\mathbb{Z}}+1\}. \] Thus even concentration means vanishing in every odd homological degree. An odd shift of the homological grading changes this condition, so an absolute normalization is necessary.

The source asks to determine both classes; it does not explicitly assert that $\mathcal P=\mathcal E$. Its original wording says "even degrees," while the supplied cleaned statement specifies "even homological degrees." The choice of grading therefore remains a source qualification requiring review. No classification or equivalence is asserted here.

Short English statement

Classify knots whose HOMFLY--PT invariant has nonnegative coefficients in the monomials $q^i(-a)^j$, and separately classify knots whose triply graded Khovanov--Rozansky homology vanishes in odd homological degrees. Both questions require fixed invariant and grading conventions, which the supplied record does not provide.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003567$, source problem code catalogue entry, titled Positivity and Evenness in HOMFLY-PT Knot Homology, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2017 and names no proposer.
• Supplied report citation: Low-dimensional Topology and Number Theory (2018), Oberwolfach Reports, pp. 2363--2426, DOI 10.4171/owr/2017/38. https://doi.org/10.4171/owr/2017/38. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: Homology of torus knots. https://doi.org/10.2140/gt.2022.26.47.
• Supplied literature reference: Triply graded link homology. https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12761.
• Supplied literature reference: Low dimensional Topology and Number Theory. https://doi.org/10.4171/owr/2017/38.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports that triply graded homology and even degree concentration are established for broad positive torus knot and link families, while positivity and the requested evenness classification for general knots remain open. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. This is a supplied editorial assessment, not independent mathematical verification of this draft.