16 problems

Let ${\cal K}$ be a topological knot type, and let a minimum-braid-index representative of ${\cal K}$ admit a negative flype. A transversally simple knot type is one for which the…

A knot is transversally simple if its transverse isotopy class is determined by its topological knot type and its Bennequin number. Positive torus knots are known to be transversal…

Self-linking classification conjecture. Negative torus knots are determined by their self-linking number; equivalently, two transversely represented negative torus knots of the sam…

For a knot or link, let the Bennequin number of a closed braid be its writhe minus its braid index. The maximal Bennequin number is the maximum of this invariant over all closed br…

Fix a smooth knot type $K$. A transverse representative $T$ of $K$ has a standard neighborhood, whose boundary has a slope, and standard neighborhoods are considered up to contact…

Let $T$ be a transverse knot in a contact manifold. A standard neighborhood of $T$ has a boundary slope, measured with respect to the chosen framing, and standard neighborhoods are…

Let $K$ be a transverse knot in $S^3$, let $\braket{\mathcal{C}_{\mathbb{Z}_p}(\Sigma_p(K),\widetilde{\xi})}$ denote the equivariant Seiberg–Witten Floer homotopy contact invariant…

Negative stabilization nonuniqueness conjecture. Negative stabilization is not a unique operation on virtual transverse knots. That is, there are two ways to negatively stabilize s…

Transverse-to-virtual injectivity conjecture. No two transverse knots are the same virtual transverse knot.

Transverse detection conjecture. If $L$ is transversely simple and $\widetilde{L}$ has the same Khovanov homology as $L$, then $\widetilde{L}$ is equivalent to $L$.

Let $m(K)$ denote the mirror image of a knot $K$. Among the 13 listed knots, namely … three knots, $m(9_{45})$, $10_{128}$, and $10_{160}$, had only been conjectured to be transver…

A transverse knot is transversely nonsimple if its transverse isotopy class is not determined by its topological knot type and self-linking number. The transverse mirror conjecture…

A transverse knot is transversely nonsimple if its transverse isotopy class is not determined by its topological knot type and self-linking number. The completeness conjecture asse…

Let $K$ be a fibered knot in $S^3$. The UTP conjecture. $K$ fails the uniform thickness property if and only if $xi_K \cong xi_{std}$, and hence if and only if $K$ is fibered stron…

Let $L$ be a link type with braid index $b>0$. For a braid $B$ whose closure represents $L$, write $n(B)$ for its braid index and $a(B)$ for its algebraic length. Kawamuro's Braid…

Let $P(-2n,-3,3)$ denote the pretzel knot with parameter $n$, and let $(L_1^n)^+$ and $(L_2^n)^-$ be the Legendrian representatives of this knot similar to those considered above.…