Journal of Knot Theory and Its Ramifications

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two $2$-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram $D$, there exists a diagram $D^{\prime }$ representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

Dean introduced twisted torus knots, which are obtained from a torus knot and a torus link by splicing them together along a number of adjacent strands of each of them. We study the knot types of these knots.

A book representation of a graph is a particular way of embedding a graph in three-dimensional space so that the vertices lie on a circle and the edges are chords on disjoint topological disks. We describe a set of operations on book representations that preserves ambient isotopy, and apply these operations to $K_6$, the complete graph with six vertices. We prove there are exactly 59 distinct book representations for $K_6$, and we identify the number and type of knotted and linked cycles in each representation. We show that book representations of $K_6$ contain between one and seven links, and up to nine knotted cycles. Furthermore, all links and cycles in a book representation of $K_6$ have crossing number at most four.

We introduce a method of constructing crosscap number two surfaces from low-complexity handcuff diagrams. This construction is then used to verify the known crosscap number two knots with crossing number twelve or less and find explicit crosscap minimizing surfaces for these knots. The collection of surfaces provides evidence that all crosscap number two knots with crossing number less than 13 have been identified, and further suggests an expansion of the satellite conjecture.

It is an open problem whether Kirk’s $\sigma$-invariant is the complete obstruction to a link map $f:S_{+}^2\cup S_{-}^2\to S^4$ being link homotopic to the trivial link. The link homotopy invariant associates to such a link map $f$ a pair $\sigma (f)=({\sigma }_{+}(f),{\sigma }_{-}(f))$, and we write $\sigma =({\sigma }_{+},{\sigma }_{-})$. With the objective of constructing counterexamples, Li proposed a link homotopy invariant $\omega =({\omega }_{+},{\omega }_{-})$ such that ${\omega }_{\pm }$ is defined on the kernel of ${\sigma }_{\pm }$ and which also obstructs link null-homotopy. We show that, when defined, the invariant ${\omega }_{\pm }$ is determined by ${\sigma }_{\mp }$, and is strictly weaker. In particular, this implies that if a link map $f$ has $\sigma (f)=(0,0)$, then after a link homotopy the self-intersections of $f(S_{+}^2)$ may be equipped with framed, immersed Whitney disks in $S^4\setminus f(S_{-}^2)$ whose interiors are disjoint from $f(S_{+}^2)$.

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean’s conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in $F$. In this paper, we extend Doleshal’s results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in $F$.

The potential function of the optimistic limit of the colored Jones polynomial and the construction of the solution of the hyperbolicity equations were defined in the authors’ previous papers. In this paper, we define the Reidemeister transformations of the potential function and the solution by the changes of them under the Reidemeister moves of the link diagram and show the explicit formulas. These two formulas enable us to see the changes of the complex volume formula under the Reidemeister moves. As an application, we can simply specify the discrete faithful representation of the link group by showing a link diagram and one geometric solution.

The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $\mathrm{\text{tr}}(K_1\#K_2)=max\{\mathrm{\text{tr}}(K_1),tr(K_2)\}$, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any width-minimizing embedding of a knot $K$ also minimizes the trunk of $K$. We produce several families of probable counterexamples to this conjecture.

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

This paper extends the work by Mednykh and Rasskazov presented in [On the structure of the canonical fundamental set for the 2-bridge link orbifolds, Universität Bielefeld, Sonderforschungsbereich 343, Discrete Structuren in der Mathematik, Preprint (1988), pp. 98–062, www.mathematik.uni-bielefeld.de/sfb343/preprints/pr98062.ps.gz]. By using their approach, we derive the Riley–Mednykh polynomial for a family of $2$-bridge knot orbifolds. As a result, we obtain explicit formulae for the volumes and Chern–Simons invariants of orbifolds and cone-manifolds on the knot with Conway’s notation $C(2n,4)$.