Narrow Results

We give a list of minimal grid diagrams of the 13 crossing prime non-alternating knots which have arc index 13. There are 9,988 prime knots with crossing number 13. Among them 4,878 are alternating and have arc index 15. Among the other non-alternating knots, 49, 399, 1,412, and 3,250 have arc index 10, 11, 12, and 13, respectively. We used the Dowker–Thistlethwaite code of the 3,250 knots provided by the program Knotscape to generate spanning trees of the corresponding knot diagrams to obtain minimal arc presentations in the form of grid diagrams.

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.

We discuss the relation between arc index, maximal Thurston–Bennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal Thurston–Bennequin number for all knots with at most 11 crossings. For some of these knots, the calculation requires a consideration of cables which also allows us to compute the maximal self-linking number for all knots with at most 11 crossings.

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.

In this paper, we define the notion of descending tangle diagrams and give a presentation of a link as a sum of two descending tangle diagrams. We also introduce a new invariant of links and find its basic properties.

We introduce the alternating tangle decomposition of a diagram of a link L and improve the upper bound of arc index α(L) by using information of the alternating tangle decomposition. Also we get the exact arc index of a class of links by combining the upper bound with Morton and Beltrami's lower bound of the arc index.

In this paper, we construct an algorithm to produce canonical grid diagrams of cable links and Whitehead doubles, which give sharper upper bounds of the arc index of them. Moreover, we determine the arc index of $2$-cable links and Whitehead doubles of all prime knots with up to eight crossings.

In this paper, we calculate the Kauffman polynomials $F(K(p,q);a,z)$ of Kanenobu knots $K(p,q)$ with $p,q$ half twists and determine their spans on the variable $a$ completely. As an application, we determine the arc index of infinitely many Kanenobu knots. In particular, we give sharper lower bounds of the arc index of $K(2n,-2n)$ by using canonical cabling algorithm and the 2-cable $\mathrm{\Gamma }$-polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations.

For the alternating knots or links, mutations do not change the arc index. In the case of nonalternating knots, some semi-alternating knots or links have this property. We mainly focus on the problem of mutation invariance of the arc index for nonalternating knots which are not semi-alternating. In this paper, we found families of infinitely many mutant pairs/triples of Montesinos knots with the same arc index.

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $\alpha (L)$ for any knot or non-split link $L$, which are $b(L)\le \frac{1}{2}c(L)+1$ and $\alpha (L)\le c(L)+2$. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of $L$.

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.

We study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants $c_1,c_2>0$ such that the ribbonlength is bounded above by $c_{1}Cr{(K)}^2$, and also by $c_{2}Cr{(K)}^{3/2}$. We use a different method for each bound. The constant $c_1$ is quite small in comparison to $c_2$, and the first bound is lower than the second for knots and links with $Cr(K)\le$ 12,748.

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.].

Recent work of Birman and Menasco discusses braid index of satellite links, and introduces a new geometrical invariant of links, which they have called the arc index. Here, we marry the work of Birman and Menasco with that of Rodolph on quasipositivity of knots, to deduce a lower bound for arc index from the Kauffman polynomial.