36 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…

A 3-strand bracket polynomial is the bracket polynomial constructed using three strands for a virtual knot diagram. Detection conjecture. The 3-strand bracket polynomial detects vi…

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…

Let $K$ be an alternating knot. Write the Links–Gould polynomial as a Laurent polynomial in $t_1,t_2$, and let its support be the lattice points corresponding to its nonzero monomi…

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…

Cable expansion conjecture. The $V_n$-polynomial of a knot is a linear combination of the $V_1$-polynomial of the knot and its $(n',1)$-cables for $n' \leq n$, with coefficients in…

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…

Let a knot have a defect defined through the zero structure of the coefficients in the differential expansion of its colored knot polynomial. Consider replacing a knot crossing by…

Let $R$ be a representation, let $\bar R$ be its conjugate, and let $R\otimes\bar R$ denote the representation product. For a knot $\mathcal K$, write $H_R^{\mathcal K}$ for its HO…

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…

Let $Z_{[33]}^{(p)}$ be the quadratic differentials associated with the rectangular representation $[33]=[3^2]$, and let $[n]$ denote the quantum number. Figure-eight [33] expansio…

Quadratic-differential expansion conjecture. For every defect-zero knot, dependence on differentials through quadratic $Z_R^{(p)}$ survives for all rectangular diagrams $R=[r^s]$.…

Let $d^{\mathcal K}$ denote the defect of the differential expansion of a knot $\mathcal K$, and let $Al^{\mathcal K}_{[1]}$ be its fundamental Alexander polynomial in topological…

Universal difference conjecture. The difference has the form

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…

Let $\mathcal{H}_{\vec{\mathsf r},\vec{\mathsf s}}(b;q,t,a)_{st}$ denote the DAHA superpolynomial expressed in standard Khovanov–Rozansky parameters, related by … with $q_{st}^2=t$…

Let $\mathcal{H}_{\{\vec{\mathsf r},\vec{\mathsf s}\}}$ be the uncolored DAHA superpolynomial associated with an algebraic knot, and write it as a polynomial in its variables. DAHA…

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…

Expansion-coefficient invariance conjecture. Transition to the quadruply graded construction does not affect the expansion coefficients $g^{\mathcal K}_{R,j}$ for all knots…

Quadruply graded reconstruction conjecture. The same change-of-variables reconstruction should hold for all two-strand torus knots and twist knots in all symmetric and antisymmetri…