Narrow Results

Every knot can be embedded in the union of finitely many half planes with a common boundary line in such a way that the portion of the knot in each half plane is a properly embedded arc. The minimal number of such half planes is called the arc index of the knot. We have identified all prime knots with arc index up to 11. We also proved that the crossing number is an upperbound of arc index for non-alternating knots. As a result the arc index is determined for prime knots up to twelve crossings.

As a supplement to the paper [Prime knots with arc index up to 11 and an upper bound of arc index for non-alternating knots, J. Knot Theory Ramifications19(12) (2010) 1655–1672], we present minimal arc presentations of the prime knots up to arc index 11.

It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of prime nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number. We also give minimal grid diagrams of some prime nonalternating knots with 13 crossings and 14 crossings whose arc index is the minimal crossing number minus one.

If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need no merge operations to deform a rectangular diagram of the trivial knot to one with no crossings.

If a rectangular diagram represents the trivial knot, then it can be deformed into the trivial rectangular diagram with only four edges by a finite sequence of merge operations and exchange operations, without increasing the number of edges, which was shown by Dynnikov in [Arc-presentations of links: Monotone simplification, Fund. Math. 190 (2006) 29–76; Recognition algorithms in knot theory, Uspekhi Mat. Nauk 58 (2003) 45–92. Translation in Russian Math. Surveys 58 (2003) 1093–1139]. Using this, Henrich and Kauffman gave in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]] an upper bound for the number of Reidemeister moves needed for unknotting a knot diagram of the trivial knot. However, exchange or merge moves on the top and bottom pairs of edges of rectangular diagrams are not considered in [Unknotting unknots, preprint (2011), arXiv:1006.4176v4 [math.GT]]. In this paper, we show that there is a rectangular diagram of the trivial knot which needs such an exchange move for being unknotted, and study upper bound of the number of Reidemeister moves needed for realizing such an exchange or merge move.

Rectangular diagrams of links are link diagrams in the plane ℝ² such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. Cromwell and Dynnikov showed that rectangular diagrams of links are useful for deciding whether a given link is split or not, and whether a given knot is trivial or not. We show in this paper that an oriented link diagram D with c(D) crossings and s(D) Seifert circles can be deformed by an ambient isotopy of ℝ² into a rectangular diagram with at most c(D) + 2s(D) vertical segments, and that, if D is connected, at most 2c(D) + 2 - w(D) vertical segments, where w(D) is a certain non-negative integer. In order to obtain these results, we show that the system of Seifert circles and arcs substituting for crossings can be deformed by an ambient isotopy of ℝ² so that Seifert circles are rectangles composed of two vertical line segments and two horizontal line segments and arcs are vertical line segments, and that we can obtain a single circle from a connected link diagram by smoothing operations at the crossings regardless of orientation.

The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentations with 13 arcs.

Let $L$ be a Montesinos link $M(-p,q,r)$ with positive rational numbers $p,q$ and $r$, each less than 1, and $c(L)$ the minimal crossing number of $L$. Herein, we construct arc presentations of $L$ with $c(L)$, $c(L)-1$ and $c(L)-2$ arcs under some conditions for $p$, $q$ and $r$. Furthermore, we determine the arc index of infinitely many Montesinos links.

In this paper, we give a list of minimal grid diagrams of the 12 crossing prime alternating knots. This is a continuation of the work in [G. T. Jin and H. J. Lee, Minimal grid diagrams of 11 crossing prime alternating knots, J. Knot Theory Ramifications29(11) (2020) Article ID: 2050076, 14 pp.].

This article explains how the authors obtained the list of prime knots of arc index not bigger than 10.