Narrow Results

We introduce an unknotting-type number of knot projections that gives an upper bound of the crosscap number of knots. We determine the set of knot projections with the unknotting-type number at most two, and this result implies classical and new results that determine the set of alternating knots with the crosscap number at most two.

A projection of the bouquet graph B with two cycles is said to be trivial if only trivial embeddings are obtained from the projection. In this paper a finite set of nontrivial embeddings of B is shown to be minimal among those which produce all nontrivial projections of B.

We show that any nontrivial reduced knot projection can be obtained from a trefoil projection by a finite sequence of half-twisted splice operations and their inverses such that the result of each step in the sequence is reduced.

In this paper we propose a region choice problem for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).

After this paper was published, the following information about doodles was pointed out by Roger Fenn. A doodle was introduced by Fenn and Taylor [2], which is a finite collection of closed curves without triple intersections on a closed oriented surface considered up to the second flat Reidemeister moves with the condition (*) that each component has no self-intersections. Khovanov [4] introduced doodle groups, and for his process, he considered doodles under a more generalized setting (i.e. removing the condition (*) and permitting the first flat Reidemeister moves). He showed [4, Theorem 2.2], a result similar to our [3, Theorem 2.2(c)]. He also pointed out that [1, Corollary 2.8.9] gives a result similar to [4, Theorem 2.2].

The authors first noticed the above results by Fenn and Khovanov via personal communication with Fenn, and therefore, the authors would like to thank Roger Fenn for these references.

An axis of a link projection is a closed curve which lies symmetrically on each region of the link projection. In this paper we define axis systems of link projections and characterize axis systems of the standard projections of twist knots.

In this paper, we introduce a distance ${\tilde{d}}_{w3}$ on the equivalence classes of spherical curves under deformations of type RI and ambient isotopies. We obtain an inequality that estimate its lower bound (Theorem 1). In Theorem 2, we show that if for a pair of spherical curves $P$ and $P^{\prime }$, ${\tilde{d}}_{w3}([P],[P^{\prime }])=1$ and $P$ and $P^{\prime }$ satisfy a certain technical condition, then $P^{\prime }$ is obtained from $P$ by a single weak RIII only. In Theorem 3, we show that if $P$ and $P^{\prime }$ satisfy other conditions, then $P^{\prime }$ is ambient isotopic to a spherical curve that is obtained from $P$ by a sequence of a particular local deformations, which realizes ${\tilde{d}}_{w3}([P],[P^{\prime }])$.

A knot $K_1$ is a minor of a knot $K_2$ if any regular projection of $K_2$ is also a regular projection of $K_1$. This defines a pre-ordering on the set of all knots. For each knot of five or less crossings, the set of all regular projections of it is determined by Taniyama [A partial order of knots, Tokyo J. Math.12(1) (1989) 205–229]. Thus, the pre-ordering is determined up to five crossing knots. In this paper, we determine the set of all regular projections of the knot $6_2$.

It is known that there exists a surjective map from the set of weak (1, 3) homotopy classes of knot projections to the set of positive knots [N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi in 2013 (Question q1). This paper obtains an answer to this question.

We have listed all families of alternating knots with the crosscap number three.

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.