Narrow Results

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n²logn) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

It has been a fundamental problem to study the crossing number and the bridge index for an 1-knot. By spinning an arbitrary 1-knot about R² in R⁴, we have a ribbon 2-knot in R⁴; therefore, for a ribbon 2-knot, we can also naturally induce notions corresponding to the crossing number and the bridge index. In this note, we will define these notions, and for a property on these notions, we will study a great contrast between 1-knots and ribbon 2-knots and in addition we will enumerate all ribbon 2-knots with the crossing number under four.

In this paper, we derive a formula that provides lower bounds on the minimal arc length required to tie a unit thickness knot in terms of the minimal crossing number of the knot. We prove that for any nontrivial knot with unit thickness, the minimal arc length is at least 24. This answers an open question (posted in [13]) in the negative: one cannot tie a knot using one foot-length of one-inch (diameter) rope. For knots with minimal crossing numbers up to 1850, our result yields larger lower bounds on the lengths of the knots than the known results.

We study the crossing number of links that are formed by edges of a triangulation of S³ with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n². In general, this bound can not be replaced by a subexponential bound. However, if is polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper bound in n for the crossing number. In our proof, we use a numerical invariant , called polytopality, introduced by the author.

For any given knot K, a thick realization K₀ of K is a knot of unit thickness which is of the same knot type as K. In this paper, we show that there exists a family of prime knots {K_n} with the property that Cr(K_n)→∞(as n→∞) such that the arc-length of any thick realization of K_n will grow at least linearly with respect to Cr(K_n).

The first aim of this paper is to give an example of a virtual knot which vanishes all the known invariants and to show that it is a non-classical (in particular, non-trivial) knot. The second is to study some properties of the Z-polynomial of a virtual knot with virtual crossing number one and to show that there are infinitely many virtual knots with virtual crossing number two.

For an arbitrary 1-knot k¹, the spun 2-knot of k¹, denoted by spun(k¹), is a ribbon 2-knot in R⁴. Hence for a ribbon 2-knot K², we can also induce a notion corresponding to the crossing number on a 1-knot, and it is said to be the crossing number of K², denoted by cr(K²). In this note, we will show that the Alexander polynomial plays an important role in determining the crossing number of a ribbon 2-knot. Lastly, we will prove the following: If k¹ is a (p,q)-torus knot, then cr(spun(k¹)) is equal to (p - 1)(q - 1).

For a knot or link K, let L(K) denote the rope length of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well-known that there exist positive constants c₁, c₂ such that for any knot or link K, c₁ · (Cr(K))^3/4 ≤ L(K) ≤ c₂ · (Cr(K))^3/2. It is also known that for any real number p such that 3/4 ≤ p ≤ 1, there exists a family of knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) = O(Cr(K_n)^p). However, it is still an open question whether there exists a family of knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) = O(Cr(K_n)^p) for some 1 < p ≤ 3/2. In this paper, we show that there are many families of prime alternating Conway algebraic knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) can grow no faster than linearly with respect to Cr(K_n).

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.

For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K) : c(K) ≤ ⌊(g(K)+9)/6⌋ and c(K) ≤ ⌊(n(K) + 16)/12⌋. The (6n - 2,3) torus knots show that these bounds are sharp.

In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link is called a predecessor of a link if and a diagram of can be obtained from a minimal diagram D of by a single crossing change. In such a case, we say that . We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

We show that for any nontrivial knot K and any natural number n, there is a diagram D of K such that the unknotting number of D is greater than or equal to n. It is well-known that twice the unknotting number of K is less than or equal to the crossing number of K minus one. We show that the equality holds only when K is a (2, p)-torus knot.

For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover, the equality holds if and only if D is an alternating diagram. For a knot K, we also estimate the minimum of d(D) + d(-D) for all diagrams D of K with c(D) = c(K).

In this paper, we prove that an adequate virtual link diagram of an adequate virtual link has minimal real crossing number.

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.

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.

Let K be a smooth knot of unit thickness embedded in the space with length L(K) and total curvature κ(K). Then where acn(K) is the average crossing number of the embedded knot K and c > 0 is a constant independent of the knot K. This relationship had been conjectured in [G. Buck and J. Simon, Total curvature and packing of knots, Topology Appl.154 (2007) 192–204] where it is shown that the square root power on the curvature is the lowest possible. In the last section we give several examples to illustrate some relationships between the three quantities average crossing number, total curvature and ropelength.

A handlebody-knot is a handlebody embedded in the 3-sphere. We enumerate all genus two handlebody-knots up to six crossings.

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, the nullification process may be the pathway for a knotted circular DNA to unknot itself (through recombination of its DNA strands). The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. It turns out that there are several different ways to define such a number, since different conditions may be applied in the nullification process. We show that these definitions are not equivalent, thus they lead to different nullification numbers for a knot/link in general, not just one single nullification number. Our aim is to explore some mathematical properties of these nullification numbers. First, we give specific examples to show that the nullification numbers we defined are different. We provide a detailed analysis of the nullification numbers for the well known 2-bridge knots and links. We also explore the relationships among the three nullification numbers, as well as their relationships with other knot invariants. Finally, we study a special class of links, namely those links whose general nullification number equals one. We show that such links exist in abundance. In fact, the number of such links with crossing number less than or equal to n grows exponentially with respect to n.

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number ([D. M. Afanasiev, Refining the invariants of virtual knots by using parity, Sb. Math.201(6) (2010) 3–18; H. A. Dye and L. H. Kauffman, Virtual crossing numbers and the arrow polynomial, in The Mathematics of Knots. Theory and Applications, eds. M. Banagl and D. Vogel (Springer-Verlag, 2010); S. Satoh and Y. Tomiyama, On the crossing number of a virtual knot, Proc. Amer. Math. Soc.140 (2012) 367–376] and so on) were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number.