20 problems

Let $K$ be an almost positive knot. Write $\Delta_K$ for its Alexander polynomial and $V_K$ for its Jones polynomial, with $\max\deg \Delta_K$ and $\min\deg V_K$ denoting the corre…

Let $A(K,f)$ be a framed link, where $K$ is its underlying link and $f$ is its framing. Let $\{K,f\}$ denote the framed-link polynomial used in the source, and let $F_K$ be the Kau…

Let $K$) be a non-trivial knot, and let $W(K)$ be a Whitehead double of $K$. The Kidwell–Stoimenow conjecture. Is it always the case that … Kidwell and Stoimenow reported that the…

Achiral-generator conjecture. Then at least one of the following holds: $K$ is achiral; $K$ is an iterated mutant of its obverse; or $K$ has self-conjugate HOMFLY and/or Kauffman p…

For a knot, let the Wada polynomial and Alexander polynomial be the knot invariants defined from representations of its braid group by automorphisms of a free group. Specialization…

Topological specialization conjecture. The two mock Alexander polynomials satisfy

Star-placement symmetry conjecture. The potentials satisfy

Assume that $q$ is not a root of unity, that $(q,t_1,t_2)$ satisfies $t_1t_2q^n=1$ for some integer $n\geq 1$, and that $(1-t_1)(1-t_2)\ne 0$. Let $Y_n$ be the $4n$-dimensional rig…

Let $q=\omega$ be a root of unity of order $N\geq 1$, let $H_{c,t}$ be the $4N$-dimensional Nichols f-algebra with scaling automorphism $\phi_t x_i=t_i x_i$, and let $R_{c,t}$ be t…

A pseudoknot $K$ has a WeRe set … where the $K_j$ are classical resolutions with weights $\alpha_j$, and let $\Delta_J$ denote the Jablan polynomial. Weighted-resolution conjecture…

Maximal conjecture. An infinite number of free parameters, namely values of $q$-dimensions for dessins that cannot be decomposed using the $[N-1]$-, $[N-2]$-, $[2]$-, $1$-, and…

Kauffman–Przytycki conjecture. For a virtual knot $L$, if a diagram of $L$ is realized in a surface of the minimal genus, then this fact is detected by the surface bracket polynomi…

Let $K=K_{r,s}$ be a torus knot with $r,s\in\mathbf{Z}$, let $R$ be a root system, and let $b\in P_+$ be a dominant weight. Define … Here $P_b$ is the symmetric Macdonald polynomia…

Primeness-detection conjecture. The Tutte polynomial and the Kauffman two-variable polynomial detect primeness: they are not factorizable for prime $KL$s.

Let $F$ be a family of knots or links, and let $L_1$ and $L_2$ be two alternating, nonisotopic $KL$s belonging to $F$. Let $P$ be a polynomial invariant. Family-distinguishing conj…

z-Parity Alexander polynomial span conjecture. The exponents satisfy

Fenn–Kauffman–Manturov conjecture. If

Let $\gamma$ be a braid on three strands, let $\hat{\gamma}$ be its closure, and let $\overline{\gamma}$ denote the braid obtained by taking the mirror operation. Let…

Let $K_1$ and $K_2$ be non-isotopic simple knots. For integers $p$ and $q$, the cable-detection conjecture asserts that there exist numbers $p$ and $q$ such that the $(p,q)$-cables…

Let $K$ be a framed long knot of unknotting number one. Let $h$ be a homotopy which unknots $K$ and satisfies … or, respectively, $ind(h)=-1$. Let $Cross(h)$ denote the crossing-ch…