Kirby Problem 1.33

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

All knots and links below are smooth in the oriented $3$-sphere, up to ambient isotopy, unless another ambient manifold is specified. A link has KR-parity if its triply graded HOMFLYPT Khovanov--Rozansky homology is supported in homological degrees of a single parity: there exists $\epsilon\in\{0,1\}$ such that $H^{i,j,k}=0$ whenever $i\not\equiv\epsilon\pmod2$. A positive link has an all-positive oriented crossing diagram. A quasipositive link is the closure of a braid that is a product of conjugates of positive Artin generators. An algebraic link is the oriented intersection of a reduced complex plane-curve singularity with a sufficiently small $3$-sphere.

Statement recorded in the supplied source.

Find topological necessary or sufficient criteria for KR-parity. In particular, (a) does KR-parity imply positivity, and does it imply quasipositivity; (b) does every algebraic link have KR-parity? [S1][S2]

Short English statement

Characterize the single-parity support of HOMFLYPT homology and test its implications for positivity and quasipositivity, as well as its occurrence for algebraic links.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2692 (KP-1.33). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3 -- A New Problem List in Low-Dimensional Topology, author preliminary edition (2026), Problem 1.33 and its remarks. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.