Journal of Knot Theory and Its Ramifications

In this paper, we present a construction of a family of surface-knot diagrams with cross-exchangeable curves, along which we can change the crossing information to obtain trivial diagrams. These diagrams also satisfy a kind of minimality, called $d$-minimal surface-knot diagrams.

We define invariants for colored oriented spatial graphs by generalizing CM invariants [F. Costantino and J. Murakami, On $SL(2,C)$ quantum $6j$-symbols and their relation to the hyperbolic volume, Quantum Topol. 4 (2013) 303–351], which were defined via non-integral highest weight representations of ${{𝒰}}_q(s{l}_2)$. We apply the same method used to define Yokota’s invariants, and we call these invariants Yokota type invariants. Then, we propose a volume conjecture of the Yokota type invariants of plane graphs, which relates to volumes of hyperbolic polyhedra corresponding to the graphs, and check it numerically for some square pyramids and pentagonal pyramids.

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

We give two criteria of links concerning a band surgery: The first one is a condition on the determinants of links which are related by a band surgery using Nakanishi’s criterion on knots with Gordian distance one. The second one is a criterion on knots with $H(2)$-Gordian distance two by using a special value of the Jones polynomial, where an $H(2)$-move is a band surgery preserving a component number. Then, we give an improved table of $H(2)$-Gordian distances between knots with up to seven crossings, where we add Zeković’s result.

An explicit formula for the $A$-polynomial of the knot with Conway’s notation $C(2n,3)$ is obtained from the explicit Riley–Mednykh polynomial of it.

For an oriented knot diagram, we define a cyclic word in letters $P$ and $N$ corresponding to positive and negative crossings along the diagram, respectively. We give a necessary and sufficient condition for a cyclic word in $P$ and $N$ to be obtained from some knot diagram. Also, we prove that any PN sequence of a diagram of a nontrivial knot contains at least four subsequences $PP$ and $NN$.

This paper has an experimental nature and contains no new theorems. We introduce certain moves for classical knot diagrams that for all examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram $D$ is a sequence of moves that transform $D$ into a diagram $D^{\prime }$ with the minimal possible number of crossings for the isotopy class of the knot represented by $D$. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain $Z_{1,2,3}$ generalizing the classical Reidemeister moves $R_{1,2,3}$, and another one $C$ (together with a variant $\tilde{C}$) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones. We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web.