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.

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.

Samuel J. Lomonaco Jr. and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco–Kauffman conjecture.

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.

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.

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.].

A petal projection of a knot $K$ is a projection of a knot which consists of single multi-crossing and non-nested loops. Since a petal projection gives a sequence of natural numbers for a given knot, the petal projection is a useful model to study knot theory. It is known that every knot has a petal projection. A petal number $p(K)$ is the minimum number of loops required to represent the knot $K$ as a petal projection. In this paper, we find the relation between a superbridge index and a petal number of an arbitrary knot. By using this relation, we find the petal number of $T_{r,s}$ as follows:

Let $r$ be an odd integer, $r\ge 3$. Then the petal number of the torus knot of type $(r,r+2)$ is equal to $2r+3$.