25 problems

Residual torsion-free nilpotence conjecture. The commutator subgroup of $\pi_1(M_J)$ is residually torsion-free nilpotent if and only if $\Delta_J$ is nontrivial.

A knot is 2-almost positive if the minimal number of negative crossings among all its diagrams is $2$. Let $L$ be a 2-almost positive diagram with even crossing number that minimiz…

Pretzel-surface weight conjecture. The nine incompressible surfaces have weights, in that order,

Let $p,q\in\mathbb{Z}$ be odd, let $n\in\mathbb{N}$, and let $K$ be the $(2n+1)$-stranded pretzel knot … For $0\leq g\leq n$, write $r_g(K)$ for the higher-genus ribbon number and…

Lecuona's conjecture. For every odd integer $a\geq 3$, the $3$-stranded pretzel knot

The $Q^r$ shift conjecture. For all odd integers $a,b,c$ and symmetric representations $[r]$,

The normalized $Q^1$ periodicity conjecture. For the indicated values of $c$ and $r$,

The $9_{46}$ transformed-factor conjecture.

The $9_{46}$ odd-index divisibility conjecture. For odd positive integers $i>1$,

The $9_{35}$ divisibility conjecture. For positive integers $i>1$,

Let $n\ge 3$, and let an even $n$-stranded pretzel knot be a pretzel knot $K=P(a_1,\dots,a_n)$ with even parameters in the sense used for pretzel knots. Write $\gamma_c(K)$ for the…

Odd pretzel strong double-slicing conjecture. The only strongly doubly slice odd pretzel links are of the form

Odd-parameter pretzel conjecture. The knot $K$ is strongly quasipositive if and only if either $|d|=n$, or $|d|=n-2$ and, after possibly reordering so that…

Let $K_j=K_j(p_1,p_2,\ldots,p_j)$ be a pretzel knot with stair-step primitive bridge spectrum. Let $T_{m,n}$ be an $(m,n)$-torus knot, and set … If…

Let $K$ be a pretzel knot. A pretzel knot is simple ribbon if a sequence of simple ribbon moves reduces it to a 1-stranded pretzel knot when it is odd, or to a 2-stranded pretzel k…

Let $K$ be a pretzel knot whose twist parameters are all greater than $1$ in absolute value. A pretzel knot is simple ribbon if a sequence of simple ribbon moves reduces it to a 1-…

Let $P(a_1,\dots,a_n)$ be the pretzel knot with $n$ odd and positive odd integers $a_1,\dots,a_n$, ordered so that $a_1\leq a_2\leq\dots\leq a_n$. Jablan–Sazdanović conjecture. … T…

Non-sliceness conjecture. For every such parameter $a$, the knot $P_a$ does not bound a disc embedded in the $4$-ball. The paper proves this when…

Let $P(p_1,\dots,p_n)$ be a pretzel knot with $|p_i|>1$ for every $i$. The paper's algorithm, given by Proposition, detects certain orderings of the parameters by exhibiting a ribb…

For odd $a\geq3$, consider the 3-stranded pretzel knots … The only 3-stranded pretzel knots for which the slice–ribbon conjecture remains open include a subset of this family. 3-st…

For integers $m\geq 0$, odd integers $a$ and $q_i$ with $a,|q_i|\geq3$ and $a\equiv1,11,37,47,49,59\pmod{60}$, define … Let $P(p_1,\dots,p_n)$ be a pretzel knot with one even param…

Let $P(p_1,\dots,p_n)$ be a pretzel knot with one even parameter, and suppose its parameters satisfy the conclusion of Theorem: after reordering, they have one of the displayed for…

Pretzel-family conjecture. The assertion proved for the $p+q=-2$ family is also true for the remaining $p+q=2,4$ cases.

Pretzel unknotting-number-one conjecture. The only 3-strand pretzel knots $P(p,q,r)$ with unknotting number one that are not 2-bridge knots are $P(3,-3,2)$ and its reflection.

Let $P(-2n,-3,3)$ denote the pretzel knot with parameter $n$, and let $(L_1^n)^+$ and $(L_2^n)^-$ be the Legendrian representatives of this knot similar to those considered above.…