Narrow Results

In this paper, we generalize virtual knot theory to multi-virtual knot theory where there are a multiplicity of virtual crossings. Each virtual crossing type can detour over the other virtual crossing types, and over classical or immersed crossings. New invariants of multi-virtual knots and links are introduced and new problems that arise are described. We show how the extensions of the Penrose coloring evaluation for trivalent plane graphs and our generalizations of this to non-planar graphs and arbitrary numbers of colors acts as a motivation for the construction of the multi-virtual theory.

In this paper a classification of Reidemeister moves, which is the most refined, is introduced. In particular, this classification distinguishes some Ω₃-moves that only differ in how the three strands that are involved in the move are ordered on the knot.

To transform knot diagrams of isotopic knots into each other one must in general use Ω₃-moves of at least two different classes. To show this, knot diagram invariants that jump only under Ω₃-moves are introduced.

Knot diagrams of isotopic knots can be connected by a sequence of Reidemeister moves of only six, out of the total of 24, classes. This result can be applied in knot theory to simplify proofs of invariance of diagrammatical knot invariants. In particular, a criterion for a function on Gauss diagrams to define a knot invariant is presented.

This paper is an exploration of simple four-regular graphs in the plane (i.e. loop-free and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921–2005), whose fascination with graphs of knots inspired this work, and with whom we had the pleasure of developing this paper.

It is well-known that for any link L, twice the unknotting number of L is less than or equal to the crossing number of L. Taniyama characterized the links which satisfy the equality. We characterize the links where twice the unknotting number is equal to the crossing number minus one. As a corollary, we show that for any link L with twice the unknotting number of L is greater than or equal to the crossing number of L minus two, every minimal diagram of L realizes the unknotting number.

Arnold introduced invariants J⁺, J^- and St for generic planar curves. It is known that both J⁺/2 + St and J^-/2 + St are invariants for generic spherical curves. Applying these invariants to underlying curves of knot diagrams, we can obtain lower bounds for the number of Reidemeister moves required for unknotting. J^- /2 + St works well to count the minimum number of unmatched RII moves. However, it works only up to a factor of two for RI moves. Let w denote the writhe for a knot diagram. We show that J^-/2 + St ± w/2 also gives sharp counts for the number of required RI moves, and demonstrate that it gives a precise estimate for a certain family of diagrams of the unknot with the underlying curve r = 2 + cos(nθ/(n + 1)), (0 ≤ θ ≤ 2(n + 1)π).

We introduce the warping polynomial of an oriented knot diagram. In this paper, we characterize the warping polynomial, and define the span of a knot to be the minimal span of the warping polynomial for all diagrams of the knot. We show that the span of a knot is one if and only if it is non-trivial and alternating, and we give an inequality between the span and the dealternating number.

We describe a correspondence between Turaev surfaces of link diagrams on S² ⊂ S³ and special Heegaard diagrams for S³ adapted to links.

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 $NN$.

An oriented monotone knot diagram is a knot diagram such that one meets each crossing as an over-crossing first as one travels the diagram with the orientation by starting at a point on the diagram. In this paper, unoriented knot projections which are monotone with an orientation and any over/under information are characterized. Also, monotone diagrams which are monotone with exactly one orientation and unique basepoint are characterized. As an application, a necessary condition for a knot projection with reductivity four is given.

The warping sum $e(K)$ of a knot $K$ is the minimal value of the sum of the warping degrees of a minimal diagram of $K$ with both orientations. In this paper, knots $K$ with $e(K)\le 3$ are characterized, and some knots $K$ with $e(K)=4$ are given.

A region crossing change at a region of a spatial-graph diagram is a transformation changing every crossing on the boundary of the region. In this paper, it is shown that every spatial graph consisting of theta-curves can be unknotted by region crossing changes.

The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure, we consider a knot diagram on the $xy$-plane in $xyz$-space missing some straight lines parallel to the $z$-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.

A knot $K$ is a parent of a knot $H$ if there exists a minimal crossing diagram $D$ of $K$ such that a subset of the crossings of $D$ can be changed to produce a diagram of $H$. A knot $K$ with crossing number $n$ is fertile if for any prime knot $H$ with crossing number less than $n$, $K$ is a parent of $H$. It is known that only $0_1,3_1,4_1,5_2,6_2,6_3,7_6$ are fertile for knots up to 10 crossings. However it is unknown whether there exist other fertile knots. A knot shadow is a diagram without over/under information at all crossings. In this paper, we introduce a definition of fertility for knot shadows. We show that if an alternating knot $K$ is fertile then the crossing number of $K$ is less than eight.

By the Tait Flyping Conjecture, all possible minimal diagrams of an alternating knot are related by a finite sequence of flypes. We use this fact for an algorithm to produce all possible minimal diagrams of an oriented alternating prime knot type. Starting with any given oriented minimal alternating diagram of a prime knot, the algorithm performs flypes exhaustively until all other possible minimal diagrams of the oriented knot type have been produced. Results for all distinct alternating prime knot types up to 16 crossings are shown. In addition, prime knot families are discussed that have either a small or a very large number of distinct minimal diagrams.

By considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu, Knot theory and game (in Japanese) (Asakura Publishing, Tokyo, 2013); L. Rudolph, Qualitative Mathematics for the Social Sciences (Routledge, London, 2013); K. Murasugi, Knot Theory and Its Applications, Translated from the 1993 Japanese original by Bohdan Kupita (Birkhauser, Boston, MA, 1996)], etc.). In this paper, we see how knots can be unknotted (and therefore how they are knotted) by considering various unknotting operations and their associated unknotting numbers.

The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of the warping degree for all oriented alternating diagrams obtained from the knot projection. In this paper, we consider the maximal number of regions which share no crossings for a knot projection with a fixed crossing, and give lower bounds for the warping degree.

Region Select is a game originally defined on a knot projection. In this paper, Region Select on an origami crease pattern is introduced and investigated. As an application, a new unlinking number associated with region crossing change is defined and discussed.

We analyze relations between several knot invariants such as the unknotting (unlinking) number, adequacy, and quasi-alternating property, and the corresponding values obtained from (minimal crossing number) knot diagrams. In particular, we define the BJ-unlinking number, and compute it for various families of knots and links for which the unlinking number is unknown.