11 problems

Let $K$ be a knot, and let $\vartheta_2(K)$ denote the characteristic-two satellite invariant. Divisibility-by-four conjecture. For every knot $K$, … The paper reports that all com…

Let $K_-$ and $K_+$ be knots such that $K_+$ is obtained from $K_-$ by a negative generalized crossing change on $2n$ strands. Characteristic-two generalized crossing-change inequa…

Let $K$ be a $\vartheta_c$-rational knot, and let $P$ be any pattern, with arbitrary winding and wrapping number. Arbitrary-pattern Rasmussen satellite conjecture. … This is propos…

A knot $K$ is $\vartheta_c$-rational when its associated tangle invariant has the form of the invariant of a rational tangle. Vanishing conjecture for $\vartheta_c$-rational knots.…

Let $K\subset S^3$ be a knot, let $c$ be either $0$ or a prime, and let $P$ be a pattern with wrapping number two and winding number zero. Write $P_t(K)$ for the $t$-twisted satell…

Let $T_{p,q}$ denote the $(p,q)$ torus knot, with $p$ and $q$ coprime positive integers. Explicit torus-knot shift conjecture. … and … This specifies the shift function in characte…

For a positive integer $q$, let $T_{p,q}$ denote the $(p,q)$ torus knot, and let $\vartheta_c$ be the invariant associated to characteristic $c$. Torus-knot shift conjecture. For e…

Let $K$ be a non-alternating knot that is the closure of a positive braid, and let $s_c$ and $\vartheta_c$ denote the corresponding Rasmussen and satellite invariants. Positive bra…

Let $\underline{s},\underline{\vartheta}\in\mathbb{Z}^{\infty}$ be infinite vectors such that, for each vector, all but finitely many entries equal the first entry, and suppose the…

Let $\mathcal{C}$ be the smooth concordance group, let $s_c$ denote the Rasmussen invariant in characteristic $c$, and let $p$ range over all primes. For a prime $p$, write…

For a knot $K$, let the $\mathfrak{sl}(N)$ Rasmussen invariants denote the Rasmussen-type invariants arising from $\mathfrak{sl}(N)$ Khovanov–Rozansky homology, and let the origina…