Related papers: A method of encoding generalized link diagrams
Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We…
We compute lower bounds on the virtual crossing number and minimal surface genus of virtual knot diagrams from the arrow polynomial. In particular, we focus on several interesting examples.
Link equivalence up to isotopy in a 3-space is the problem that lies at the root of knot theory, and is important in 3-dimensional topology and geometry. We consider its restriction to alternating links, given by two alternating diagrams…
Binary representations of the trefoil and other knots of up to ten crossings in the simple cubic lattice were created. The BiEntropy of each knot was computed using a variety of binary encodings and compared against controls. This showed…
The problem of serving multicast flows in a crossbar switch is considered. Intra-flow linear network coding is shown to achieve a larger rate region than the case without coding. A traffic pattern is presented which is achievable with…
We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots…
Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A multi-crossing is a crossing where more than two strands meet at a single point, such that each strand bisects the…
Shimizu introduced a region crossing change unknotting operation for knot diagrams. As extensions, two integral region choice problems were proposed and the existences of solutions of the problems were shown for all non-trivial knot…
The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm…
This is an expository article on diagrammatic representations of knots and links in various settings via braids.
In the previous paper, we considered a link diagram invariant of Hass and Nowik type using regular smoothing and unknotting number, to estimate the number of Reidemeister moves needed for unlinking. In this paper, we introduce a new link…
Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results.…
This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a…
Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R.…
For each link type $K$ in the 3-sphere, we show that there is a polynomial $p_K$ such that any two diagrams of $K$ with $c_1$ and $c_2$ crossings differ by at most $p_K(c_1) + p_K(c_2)$ Reidemeister moves. As a consequence, the problem of…
A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs…
Every link in R^3 can be represented by a one-vertex ribbon graph. We prove a Markov type theorem on this subset of link diagrams.
We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We…
The scarcity of qubits is a major obstacle to the practical usage of quantum computers in the near future. To circumvent this problem, various circuit knitting techniques have been developed to partition large quantum circuits into…
We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded…