Kirby Problem 1.24

Definitions and mathematical statement

Let $U$ be the unknot and let $K\subset S^3$ be a smooth knot. Normalize the Jones polynomial by $V_U(t)=1$ and the oriented skein relation \[ t^{-1}V_{L_+}(t)-tV_{L_-}(t) =(t^{1/2}-t^{-1/2})V_{L_0}(t). \] The three links agree outside a ball containing a positive crossing, a negative crossing, or their oriented smoothing. Normalize the HOMFLYPT polynomial by $P_U(a,z)=1$ and $aP_{L_+}-a^{-1}P_{L_-}=zP_{L_0}$. Let $J_n(K;q)$ be the zero-framed $\mathfrak{sl}_2$ colored Jones invariant, with the knot colored by the $n$-dimensional irreducible representation and normalized to $J_n(U;q)=1$.

Does a nontrivial knot exist satisfying, respectively,

1. $V_K(t)=1$;
2. $J_n(K;q)=1$ for every integer $n\geq1$;
3. $P_K(a,z)=1$?

These are polynomial identities in formal variables, not equalities at a selected numerical evaluation. In part (b) a single knot must satisfy all colors simultaneously. Each negative answer would say that the indicated invariant detects the unknot among knots; the restriction to one component is essential.

Short English statement

Can a nontrivial knot have the unknot's Jones polynomial, every colored Jones polynomial, or HOMFLYPT polynomial?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2683 (KP-1.24), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 1.24. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.