3 problems

Let $\hat\beta_1$ and $\hat\beta_2$ be positive braid knots. A positive path from $\hat\beta_2$ to $\hat\beta_1$ is a sequence of positive braid knots beginning at $\hat\beta_2$ an…

Let $T(2,a)$ and $T(4,b)$ denote torus knots, and write $K\leq_g J$ when $K$ is Gordian adjacent to $J$, meaning that $K$ can be obtained from $J$ by changing exactly $u(J)-u(K)$ c…

Let $T(p,q)$ and $T(r,s)$ denote torus knots, with their indices given by the first parameters. The smaller-index obstruction conjecture. The torus knots $T(3,4)$ and $T(3,5)$ cann…