99 problems

Given a knot $K$, let the smooth non-orientable four-genus be … with $\gamma_4(K)=0$ when $K$ is slice. For a torus knot $T_{p,q}$, let $\vartheta(T_{p,q})$ denote the number of pi…

Let $K$ be a torus knot, and let $\gamma_4(K)$ denote its smooth 4-genus. A realizable pair $(e,h)$ records the normal Euler number and first Betti number of a smooth surface in…

Let $T_{n,s}$ denote the torus knot of type $(n,s)$, where $n$ and $s$ are relatively prime integers with $2\le n<s$. The petal number $p(K)$ is the minimum number of loops among p…

A knot diagram's unknotting number $u(K)$ is the minimum number of crossing changes, over all diagrams of a knot $K$, needed to transform $K$ into the unknot. For coprime positive…

A knot is exchange reducible if every closed braid representative can be reduced to a minimal braid representative using braid isotopy, exchange moves, and destabilizations. Torus…

A knot is transversally simple if its transverse isotopy class is determined by its topological knot type and its Bennequin number. Positive torus knots are known to be transversal…

Self-linking classification conjecture. Negative torus knots are determined by their self-linking number; equivalently, two transversely represented negative torus knots of the sam…

For an integer $r>1$ and a knot $K$, let $\mathrm{mincol}_r K$ denote the minimum number of colors required for a non-trivial $r$-coloring, minimized over all diagrams of $K$. For…

For each $n$, let $\mathcal H_s(T_n)$ be a complex with Poincaré polynomial $\mathcal P_s(T_n)$ and differentials $d_N$. Let “smallest” mean minimal with respect to the properties…

For positive integers $m,n$, define the normalized reduced superpolynomial … When the limit exists, call $\mathcal P_s(T_n)=\lim_{m\to\infty}\mathcal P_s(T_{m,n})$ the stable super…

Let $T_{3,m}$ be a torus knot, and let $\mathcal P(T_{3,m})$ denote its reduced superpolynomial. For $m=3k+1$ or $m=3k+2$, define the three polynomials…

Hull number conjecture for the $(3,4)$-torus knot. The $(3,4)$-torus knot has hull number $3$.

Let $K_{8,3}$ be the $(8,3)$ torus knot, which the paper describes as forming an eight-sided regular polygon without a central gap. Its Ribbonlength is the minimum length-to-width…

Let $q$ be a positive integer and let $K_{q+1,q}$ denote the $(q+1,q)$ torus knot. The closed Ribbonlength is the minimum length-to-width ratio among closed ribbon presentations. T…

Let $K_{4,3}$ be the $(4,3)$ torus knot, and let Ribbonlength mean the minimum length-to-width ratio among ribbon presentations of a knot. The construction in the paper has ratio…

Let $q$ be a positive integer and let $K_{q+1,q}$ denote the $(q+1,q)$ torus knot. Its closed Ribbonlength conjecture is … The conjecture concerns the minimum length-to-width ratio…

Let $T(p,q)$ be a torus knot of type $(p,q)$, where $p$ and $q$ are coprime integers. Let $\rho_{j,k}\colon \pi T(p,q)\to\operatorname{SO}(3)$ send the standard generators $a$ and…

Existence problem. Is there a torus knot that is not contained in $\mathcal T$?

Non-exceptional torus knot conjecture. No non-exceptional torus knot belongs to $\mathcal T$.

Let $M$ be a $3$-manifold, let $g$ be any non-negative integer, and consider the irreducible representations of the fundamental group of $M$ together with their adjoint Reidemeiste…

Let $K$ be a knot in $S^3$. Its normalized recurrence polynomial is $\alpha_K(t,M,L)$, and let $A_K(M,L)$ denote its $A$-polynomial. Remove all repeated irreducible factors involvi…

Let $p,q>0$ be coprime integers with $pq eq 0pmod{2}$, and let $mathcal{T}{(p,q)}$ be a $1$-efficient ideal triangulation of the complement of the $(p,q)$ torus knot. The alternati…

A torus knot is a knot $T(s,t)\subseteq S^3$ with $\gcd(s,t)=1$. Let Proposition 5.2 denote the proposition preceding this claim, concerning the chamber parameters and associated c…

A torus knot is a knot $T(s,t)\subseteq S^3$ with $\gcd(s,t)=1$. Let Proposition 1.3 denote the proposition preceding this conjecture, which establishes the stated well-defined Lau…

Let $T_{p,q}$ be the $(p,q)$-torus knot, where $p$ and $q$ are relatively prime integers, and let ${\rm u}^u_b(T_{p,q})$ denote its unoriented band unknotting number, the minimum n…