Problem 1.12 — Categorify other knot polynomials.

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.

A knot or link is a smooth oriented embedding of one or more circles in the oriented $3$-sphere, considered up to orientation-preserving ambient isotopy. Write $O$ for the unknot, which bounds an embedded disk. Categorifying a polynomial invariant means assigning graded homology groups, invariant under the relevant knot or link equivalence, whose graded Euler characteristic recovers that polynomial. For a multigrading $\mathbf j$ and homological degree $i$, this characteristic is $\sum_{i,\mathbf j}(-1)^i\dim H^{i,\mathbf j}\,\mathbf x^{\mathbf j}$, whenever the finiteness conditions make it defined.

Statement recorded in the supplied source.

Construct meaningful categorifications of knot polynomials other than the Jones polynomial, including appropriate graded chain complexes, equivalence-invariant homology and a proof that the graded Euler characteristic is the target polynomial. Seek actual homological constructions with structural information, rather than assigning vector spaces solely from known coefficients. [S2]

Short English statement

Construct invariant homology theories whose graded Euler characteristics recover other knot polynomials.

Sources

• S1 ULAM. AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10400012 (AMR-103-0012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 T. Ohtsuki (editor), Problems on invariants of knots and 3-manifolds, Geometry & Topology Monographs 4 (2002), pp. 377--572 (published 2004), item 1.12 and its surrounding definitions and remarks. https://arxiv.org/abs/math/0406190v1.