19 problems

Let $K$ be a knot, let $\overline{K}$ denote its mirror image, and let $-\overline{K}$ denote the mirror image with reversed orientation. The Kauffman polynomial of $K$ is denoted…

NP versus P conjecture. $NP\neq P$.

Let $K$ be a knot, let $W_K$ be its Whitehead double, and let $F(K)$ be the Kauffman polynomial of $K$ in the variables $a$ and $z$. Stoimenow–Kidwell conjecture. Then … The questi…

For each knot $K$, consider a triply graded homology theory with graded pieces $\mathcal H^{\rm Kauffman}_{i,j,k}(K)$, and let $\{d_N\}$ be a family of differentials together with…

Let $K$ be a knot, and let $\mathcal F(K)$ denote a finite reduced polynomial in $\mathbb{Z}[\lambda^{\pm 1},q^{\pm 1},t^{\pm 1}]$. Let $\operatorname{HSO}_N(K;q,t)$ and…

Let $K$ be a knot. A finite polynomial $\overline{\mathcal F}(K)$ in $\mathbb{Z}[\lambda^{\pm 1},q^{\pm 1},t^{\pm 1}]$ is sought, with specializations to the $so(N)$ and $sp(N)$ kn…

Corrected HZ factorisability conjecture. If the HOMFLY–PT/Kauffman relation holds, then the HZ transform is factorisable:

Petrou's HZ factorisability conjecture. The HOMFLY–PT/Kauffman relation holds if and only if the HZ transform is factorisable:

Kauffman-polynomial span conjecture. The spans satisfy

HOMFLY–PT–Kauffman criterion. HZ factorisation for knots occurs if and only if

Let $L$ be a solid torus link with wrapping number $\operatorname{wr}$, and let $\nabla_L$ denote its Kauffman polynomial. Wrapping-degree equality conjecture. … Equality holds for…

Let $\operatorname{BMW}^{\mathfrak{Leg}}_n$ be the Legendrian BMW algebra and let $\operatorname{BMW}^{\infty}_n$ be the specialized BMW algebra, with…

Let $\mathcal{C}$ be a braided fusion category with Property F, and let its associated link invariant be the invariant obtained from the braid-group representations of…

Let ${\mathcal L}$ be a non-oriented link with $L$ components, let ${\mathcal G}^{(c)}$ and ${\mathcal H}^{(c)}$ denote the connected Kauffman and HOMFLY contributions in the funda…

Let a Kauffman invariant of a knot or link be compared with the corresponding HOMFLY contribution of $U(N)$. A composite representation of $U(N)$ is a representation denoted…

Let ${\cal L}$ be a two-component link whose components are both colored by the fundamental representation, denoted by $\tableau{1}$. Let $g_{\tableau{1},\tableau{1}}({\cal L})$ be…

Let ${\cal K}$ be the unknot, and let $\hat h_R$ and $\hat g_R$ denote its reformulated colored Kauffman invariants. Unknot vanishing conjecture. Apart from the representations alr…

Let ${\cal L}$ be an unoriented link with $L$ components, colored by representations $R_1,\ldots,R_L$. Let $\hat h_{R_1,\ldots,R_L}(t,\nu)$ and $\hat g_{R_1,\ldots,R_L}(t,\nu)$ be…

Let ${\cal K}$ be a knot, let $R$ be a representation, and let $\hat h_R(t,\nu)$ and $\hat g_R(t,\nu)$ be the reformulated colored Kauffman invariants obtained by applying the inve…