13 problems

Let $K$ be a knot, let $D^1(K)$ denote its untwisted positive Whitehead double, and let $\chi_s$ denote the slice Euler characteristic. Slice Euler characteristic conjecture. … Thi…

Let $K$ be a knot, let $u_s(K)$ denote its slicing number, and let $g_s(K)$ denote its slice genus. Unbounded difference conjecture. The difference … can be arbitrarily large. This…

A slice-torus invariant is a real-valued function $f$ on the smooth knot concordance group satisfying, for knots $K,K'$ in $S^3$, $|f(K)|\leq g_4(K)$,…

Let $DTS^2$ be the unit disk bundle of the tangent bundle of $S^2$, with boundary $\mathbb{RP}^3$. Let $(\Sigma,K)\subset(DTS^2,\mathbb{RP}^3)$ be a properly embedded orientable co…

Let $K_m$ denote a twist knot with $m$ twists, and suppose that $K_m$ is non-slice. The $4$-dimensional clasp number is $1$ for every such knot, and the $T$-genus should grow linea…

Let $P$ be a pattern, meaning a knot in a solid torus, and let $K$ be a knot in $S^3$. Write $P(K)$ for the satellite knot and $P(U)$ for the satellite of the unknot, and let…

Dye–Kaestner–Kauffman conjecture. For every classical knot $K$,

Vanishing conjecture. If $K$ is slice, then

Let $K$ be a knot, let $g_4(K)$ denote its smooth orientable slice genus, and let $b3_4(K)$ denote its smooth non-orientable slice genus. A knot is non-slice if $g_4(K)>0$. Murakam…

Sliceness conjecture for 2-bridge knots. For all $2$-bridge knots $K$, $g_{4}(K)=0$ if and only if $g_{4}^{{\rm top}}(K)=0$.

Let $K$ be a knot, and let $s_n(K)\in 2\mathbb{Z}$ denote the even integer determined by the Gornik homology of $K$ for each $n\geq 2$. Divisibility conjecture for the Gornik invar…

Let $K$ be a knot, and let $s_n(K)\in 2\mathbb{Z}$ denote the even integer determined by the Gornik homology of $K$ for each $n\geq 2$. Lobb's proportionality conjecture. For any k…

Let $U_s$ denote the relevant unknotting invariant and let $g_s$ denote the slice genus. Livingston's conjecture. The difference $U_s-g_s$ can be arbitrarily large. This predicts a…