38 problems

Let $D_{p,p}$ and $D_{p,p+1}$ be the torus-knot diagrams used in the paper, and let $H^{i,j}(D)$ denote their Khovanov homology in homological degree $i$ and quantum degree $j$. No…

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…

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…

Let $p,q>0$ be coprime integers with $pq eq 0\pmod{2}$, 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 $n,m$ be coprime positive integers, and let … For a positive integer $r$, write … for the irreducible representation locus, and let…

Torus-knot dimer-model conjecture. Every $(p,q)$ torus knot admits a dimer model.

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…

For an odd positive integer $n$, consider a torus knot $T(2,n)$ and the pairs $(e,h)$ of normal Euler number and first Betti number of smooth surfaces in $B^4$ bounded by $T(2,n)$.…

Let $m,n$ be coprime positive integers, let $C_{\frac{m}{n}}$ be the closure of $y^m=x^n$ in $\mathbb{P}^2$, and let $J_{\frac{m}{n}}$ be its compactified Jacobian. Let $P$ be the…

Let $m,n$ be coprime positive integers, let $\mathfrak{h}$ be the $(n-1)$-dimensional standard representation of $S_n$, and let $\mathrm{L}_{\frac{m}{n}}$ be the unique finite-dime…

Let $K_n(t,q)$ denote the Poincaré polynomial of the Khovanov homology of the torus knot $T(n+1,n)$. Shumakovitch–Turner recurrence conjecture. The polynomials satisfy … and, for…

Allen's conjecture. If a connected sum of possibly several torus knots is concordant to an $L$-space knot, then it is concordant to a positive torus knot.

Let $r$ and $s$ be relatively prime integers with $1 < r < s$, and let $T_{r,s}$ denote the corresponding torus knot. The petal-number bound for torus knots. … The authors are led…

Let $2\leq p<p'$ be coprime positive integers, let $2\leq r$ with $p<r$, and let $T(p,p')$ be the $(p,p')$ torus knot. Let $L_r(n\Lambda_1)$ denote the irreducible…

Let $(p,q)$ be a pair of coprime integers specifying a torus knot, and consider an overtwisted contact structure that supports non-loose Legendrian $(p,q)$--torus knots. The mounta…

Classification conjecture. The only non-orientably fillable torus knots are of the form $T(p,2)$ with $p<0$.

Let $T_{p,q}$ be the $(p,q)$-torus knot in $S^3$, where $p$ and $q$ are positive coprime integers, and let $K$ be a knot concordant to $T_{p,q}$. A ribbon concordance is a concorda…

Torus-knot slope-jump conjecture. For the torus knot $T_{n,n+1}$, the jump in the slope $\Delta \Phi'_{T_{n,n+1}}(\alpha)$ is equal to $0$ except when $\alpha=\frac{2+2k}{2+3k}$ fo…

Let $K$ be a torus knot, let $\gamma_4(K)$ denote its smooth nonorientable 4-genus, and let $(e,h)$ denote the normal Euler number and nonorientable genus of a surface bounded by…

For an odd integer $n$, let $T(2,n)$ denote the corresponding torus knot, and let $(e,h)$ denote the pair consisting of the normal Euler number and nonorientable genus of a surface…

Torus-knot alternating-concordance conjecture. A sum of torus knots is concordant to an alternating knot if and only if it is a sum of $(2n+1,2)$ torus knots.

Concordance conjecture for sums of torus knots. If a connected sum of possibly several torus knots is concordant to an $L$-space knot, then it is concordant to a positive torus kno…

Let $T_{p,q}$ denote the torus knot associated to positive coprime integers $p$ and $q$, and let $\widehat{HFK}(T_{p,q})$ be its knot Floer homology. Write…