Journal of Knot Theory and Its Ramifications

We construct a three-variable polynomial invariant for unoriented kd($1$)-linkoid diagrams including knotoid ones as a certain weighted sum of polynomials on oriented kd($1$)-linkoid diagrams associated with a given kd($1$)-linkoid diagram.

In this paper, we construct a polynomial invariant of Kauffman type for kd($1$)-linkoids and compute the polynomials for knotoids with up to three crossings. As a consequence, it is shown that the polynomial is different from the previous one in [Y. Miyazawa, A polynomial invariant of Kauffman type for knotoids, preprint].

Experimental data from Dunfield et al. using random grid diagrams suggest that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model of all 2-bridge knots of a given crossing number, almost all counted twice. We present a convenient way to count Seifert circles in this model and use this to compute a lower bound for the average Seifert genus of a 2-bridge knot of a given crossing number.

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston’s gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a boundary-parabolic $\mathrm{\text{PSL}}(2,ℂ)$-representation to boundary-unipotent $\mathrm{\text{SL}}(2,ℂ)$-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.

In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an $S$-homotopy class of a nanoword into the $S$-homotopy class of the empty word. In this paper, we study the relation between the swap move of letters and the forbidden moves of virtual diagrams, that is, local moves of virtual diagrams which unknot any virtual knot.

An unknotting operation is a local move such that any knot diagram can be transformed into a diagram of the trivial knot by a finite sequence of these operations plus some Reidemeister moves. It is known that for all $n\ge 2$ the $H(n)$-move is an unknotting operation for classical knots and links. In this paper, we extend the classical unknotting operation $H(n)$-move to virtual knots and links. Virtualization and forbidden move are well-known unknotting operations for virtual knots and links. We also show that virtualization and forbidden move can be realized by a finite sequence of generalized Reidemeister moves and $H(n)$-moves.