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Let r be an odd integer, r ≥ 3. Then the petal number of the torus knot of type (r , r + 2) is equal to 2 r + 3. [ABSTRACT FROM AUTHOR]

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.]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Lomonaco and Kauffman introduced knot mosaics in their work on quantum knots. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. In this paper, we consider the alternating mosaic number of an alternating knot K which is defined as the smallest integer n for which K is representable as a reduced alternating knot n -mosaic. We define a signed mosaic graph and a diagonal grid graph and construct Hamiltonian cycles derived from the diagonal grid graphs. Using the cycles, we completely determine the alternating mosaic number of torus knots of type (2 , q) for q ≥ 2 , which grows in an order of q 1 / 2. [ABSTRACT FROM AUTHOR]

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell, and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh (and, independently, Barsotti and Mattman) proved there are exactly 14 intrinsically knotted graphs with 21 edges by showing that H 12 and C 14 are the only triangle-free intrinsically knotted graphs of size 21. Our current goal is to find the complete set of intrinsically knotted graphs with 22 edges. To this end, using the main argument in [9] , we seek triangle-free intrinsically knotted graphs. In this paper we present a new intrinsically knotted graph with 22 edges, called M 11 . We also show that there are exactly three triangle-free intrinsically knotted graphs of size 22 among graphs having at least two vertices with degree 5: cousins 94 and 110 of the E 9 + e family, and M 11 . Furthermore, there is no triangle-free intrinsically knotted graph with 22 edges that has a vertex with degree larger than 5. [ABSTRACT FROM AUTHOR]

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on 'Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot -mosaic is an matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot -mosaics for any positive integers and , toroidal knot -mosaics for co-prime integers and , and furthermore toroidal knot -mosaics for a prime number . We also analyze the asymptotics of the growth rates of their cardinality. [ABSTRACT FROM AUTHOR]

In this paper, we calculate the Kauffman polynomials of Kanenobu knots with half twists and determine their spans on the variable 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 by using canonical cabling algorithm and the 2-cable -polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations. [ABSTRACT FROM AUTHOR]

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 -cable links and Whitehead doubles of all prime knots with up to eight crossings. [ABSTRACT FROM AUTHOR]

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except link, then m(K) ≤ c(K) - 1. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]