16 problems

Fourth skein module generation conjecture. The fourth skein module $\mathcal{S}_4(S^3)$ is generated by trivial links; the fourth skein module of $2$-tangles in a disk is generated…

Montesinos–Nakanishi conjecture. Every link is $3$-equivalent to a trivial link. More generally, every $2$-tangle is $3$-equivalent to a tangle with at most one crossing; every $3$…

Let $M$ be a matroid with a tangle $\mathcal{T}$ of order $k\geq 4$ and breadth $m$. The half-breadth internally 4-connected minor conjecture. The matroid $M$ has an internally $4$…

Let $M$ be a matroid with a tangle $\mathcal{T}$ of order $k\geq 4$. For each $i\in\{4,5,\ldots,k\}$, let $T_i(\mathcal{T})$ be the truncation of $\mathcal{T}$ to order $i$. The si…

Define a sequence of non-negative integers by $t_0=0$ and $t_i=3t_{i-1}+1$ for $i\geq 1$, equivalently $t_i=(3^i-1)/2$. Let $M$ be a matroid with a breadth-critical tangle…

Let $k\geq 2$ and let $M$ be a matroid with a breadth-critical tangle $\mathcal{T}$ of order at least $k$. A matroid is $(s_0,s_1,\ldots,s_t)$-connected if, whenever…

Let $T$ be a framed oriented $r$-component tangle in ${\mathbb{R}}^3_{x\geq 0}$, let $N(\varLambda_T)$ be its specified neighborhood, and let $H$ denote the reference subspace used…

Let $T$ be a Conway tangle with the specified connectivity and no closed components, and let $\{K_n\}_{n\in\mathbb{Z}}$ be the associated family of knots. Say that $T$ is horizonta…

Let $T$ be a Conway tangle, and let $K_0$ and $K_1$ be the knots obtained from the associated family of tangle fillings. Say that $T$ is horizontally split when it admits the corre…

Let $T$ be a tangle. Write $\operatorname{ThinHF}(T)$ and $\operatorname{ThinKh}(T)$ for the spaces of thin fillings associated respectively to Heegaard Floer homology and Khovanov…

Decomposition criterion. The graphs $\Gamma$ and $\Gamma'$ are equivalent if and only if they admit decompositions

A free tangle is a tangle whose crossings are free crossings, and its four anchor points remain fixed on the boundary. The resultant knot probability is the proportion of crossing…

A free $21n$ knot is the free knot diagram formed from the $21$-tangle and a free $n$-tangle, as used in the source. Resultants are knots obtained by assigning all free crossings.…

Irreducible tangles are persistent conjecture. Every irreducible tangle is a persistent tangle.

Multi-separation star refinement conjecture. There exists a set of separations $\{\overleftarrow{u}_i:i\in[n]\}$ such that:

Canonical sequence conjecture. For every graph $G$ there exists a canonical sequence of tree-decompositions $(T_i,\mathcal{V}_i)$ for $1 \leq i \leq n$ of $G$ such that: