16 problems

Mirror-pair conjecture. These $56$ knotoids form $28$ mirror pairs. This is supported by the observed mirror-pair structure of the final reduction, but the source presents it as a…

Let $K_i$ be a knotoid diagram, and let $\operatorname{Reid}_r(K_i)$ denote the set of diagrams obtained by systematically performing Reidemeister moves with $r\geq 0$ crossing-inc…

Region-distinction conjecture. For every positive integer $n$, there is a planar knotoid diagram $K_n$ with exterior region $R_0$ at infinity such that

Spherical-achirality conjecture. There are infinitely many spherically achiral knotoids that are planarily chiral.

Let $K$ be a knotoid diagram. Let $K_1$ and $K_2$ be the starred knotoid diagrams obtained from $K$ by placing a star in the region incident to the tail and the region incident to…

Topological specialization conjecture. The two mock Alexander polynomials satisfy

Star-placement symmetry conjecture. The potentials satisfy

Minimum-volume conjecture. The unique spherical knotoid of smallest volume is the knotoid $2_1$, and its volume is approximately $5.33349\dots$.

Let $K_1$ and $K_2$ be prime planar knotoids with $5$ crossings. Write $K_1\simeq K_2$ when they are equivalent as planar knotoids. Goundaroulis et al.'s distinctness conjecture. T…

Let $D$ be a linear chord diagram with one chord for a planar knotoid, and write its corresponding word as $c=a^{n_1}b^{m_1}\dots a^{n_k}b^{m_k}$, where $k<\infty$ and…

A spherical knotoid is a knotoid in $S^2$, and its crossing number is denoted by $n$. Let $\|CH^+(K)\|$ and $\|CH^-(K)\|$ denote the numbers of positive and negative chord pairs as…

Let $K$ be a virtual knot of genus one, and let a torus representation be a representation of $K$ by a knot diagram in the thickened torus. The surface bracket states of such a rep…

Let $s\kappa_n$ be the number of $n$-crossing knotoid diagrams, let $sk_n$ be the number of $n$-crossing knotoid shadows, and let $\mu_K$ be the knotoid-shadow connective constant.…

Let $sk_n$ be the number of $n$-crossing knotoid shadows, let $\mu_I$ be the connective constant for knot shadows, and let $\mu_K$ be the connective constant for knotoid shadows. E…

Classical knotoid embedding conjecture. If two knotoid diagrams in $S^2$ are virtually equivalent, then they are equivalent to each other by finitely many $\Omega$-moves and isotop…

Crossing-number conjecture. Every minimal diagram of the knotoid $\kappa^\bullet$ has complexity $0$.