8 problems

For a spiral knot $S(p,q,\varepsilon)$, let $b(S(p,q,\varepsilon))$ denote its bridge number, and let $\min(p,q)$ be the minimum of its two integer parameters. Bridge-number conjec…

Let $n$ be a positive integer, and let $\{\mathfrak{n},\ldots,1,0\}$ be a sequence. Link spectrum realization conjecture. Any sequence $\{\mathfrak{n},\ldots,1,0\}$ satisfying a se…

A bridge-braid spectrum is a sequence of integers associated to a knot, with the entries decreasing and ending in $\{\ldots,1,0\}$. Bridge-braid spectrum realization conjecture. Fo…

Let $b(d)$ denote the entries of the bridge-braid spectrum of a knot, indexed by the relevant integer parameter $d$. Concavity conjecture. The sequence $\{b(d)\}$ is concave upward…

Tie-knot conjecture. All tie knots are alternating 2-bridge knots.

Minimal-diagram bridge-number realization conjecture. For any knot $K$,

Let $L$ be a link, let $\omega(L)$ denote its Wirtinger number, and call $L$ to have Property M if $\omega(L)$ is realized by a minimum-crossing diagram of $L$. For a knot diagram…

Kosaka's conjecture. Virtual knots have the same virtual bridge number if they have the same primary data.