11 problems

Let $K$) be a knot, let $D(K)$ denote its positively-clasped Whitehead double, and let $[K]$ and $[D(K)]$ denote their elements in the knot concordance group $\mathcal{C}$. Kirby’s…

Let $K$ be a knot, let $c(K)$ denote its crossing number, and let $W(K)$ be its Whitehead double. Tripp's conjecture. The crossing number of any knot coincides with the canonical g…

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 non-trivial knot, and let $W(K)$ be a Whitehead double of $K$. The Kidwell–Stoimenow conjecture. Is it always the case that … Kidwell and Stoimenow reported that the…

Let $K$ be a knot, let $W_K$ be its Whitehead double, and let $F(K)$ be the Kauffman polynomial of $K$ in the variables $a$ and $z$. Stoimenow–Kidwell conjecture. Then … The questi…

Let $\mathcal{C}$ be the knot concordance group, and let $D$ send a knot-concordance class to the class of its positively-clasped Whitehead double. Hedden–Kirk linear-independence…

Let $s$ denote Rasmussen's invariant, and let $t_{s/2}$ be the integer-valued invariant obtained from the slice-torus invariant $s/2$ by taking the maximal twisting parameter at wh…

Let $K$ be a prime knot, let $c(K)$ denote its crossing number, and let $W(K)$ be its Whitehead double. Jang–Lee's conjecture. The crossing number of any prime knot coincides with…

Let $T_{2,3}$ be the $(2,3)$ torus knot, and let $D(K)$ denote its positively-clasped untwisted Whitehead double. Two knots are smoothly concordant when their connected sum with on…

For a knot $K$, let its untwisted Whitehead double be the satellite knot obtained using the untwisted Whitehead pattern. Untwisted Whitehead-double conjecture. A knot is slice if a…

A knot in the 3-sphere is slice if it is concordant to the unknot, equivalently if it bounds a smooth properly embedded disk in the 4-ball. For a knot $K$, let $D(K)$ denote its (p…