Journal of Knot Theory and Its Ramifications

We note a simple algebraic proof of Frolkina’s result that ${ℝ}^3$ does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions.

We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.

For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When $n\le 5$, the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can parametrize the space of equilateral hexagons with a set of measure preserving action-angle coordinates. With this coordinate system, we provide new bounds on the knotting probability of equilateral hexagons.

Let $K,K^{\prime }$ be ribbon knottings of $n$-spheres or tori in $S^{n+2}$, $n\ge 2$. We show that if the knot quandles of these knots are isomorphic, then the ribbon knottings are stably equivalent, in the sense of Nakanishi and Nakagawa, after taking a finite number of connected sums with trivially embedded copies of $S^{n-1}\times S^1$.

We define a category $v{𝒯}$ of tangles diagrams drawn on surfaces with boundaries. On the one hand, we show that there is a natural functor from the category of virtual tangles to $v{𝒯}$ which induces an equivalence of categories. On the other hand, we show that $v{𝒯}$ is universal among ribbon categories equipped with a strong monoidal functor to a symmetric monoidal category. This is a generalization of the Shum–Reshetikhin–Turaev theorem characterizing the category of ordinary tangles as the free ribbon category. This gives a straightforward proof that all quantum invariants of links extend to framed oriented virtual links. This also provides a clear explanation of the relation between virtual tangles and Etingof–Kazhdan formalism suggested by Bar-Natan. We prove a similar statement for virtual braids, and discuss the relation between our category and knotted trivalent graphs.

By using the HOMFLY skein theory, we prove a strong integrality theorem for the reduced colored HOMFLYPT invariants defined by a basis in the full HOMFLY skein of the annulus.

The minimal coloring number of a $\mathbb{Z}$-colorable link is defined as the minimum number of colors for nontrivial $\mathbb{Z}$-colorings on diagrams of the link. In this paper, we show that the minimal coloring number of any nonsplittable $\mathbb{Z}$-colorable links is four.