Lattice stick number sL(K) is defined to be the minimal number of sticks required to construct a polygonal representation of the knot K in the cubic lattice. In this paper, we give lattice stick numbers of small knots such as 31 and 41. More precisely we prove that sL(31) = 12 and sL(K) ≥ 14 for any other non-trivial knot K.
Lomonaco and Kauffman introduced a knot mosaic system to give a definition of a quantum knot system which can be viewed as a blueprint for the construction of 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 by adjoining properly that is called suitably connected. Dn denotes the total number of all knot n-mosaics. Already known is that D 1 = 1, D 2 = 2, and D 3 = 22. In this paper we establish the lower and upper bounds on Dn 2 275 (9 · 6 n−2 + 1) 2 · 2 (n−3) 2 ≤ Dn ≤ 2 275 (9 · 6 n−2 + 1) 2 · (4.4) (n−3) 2 .and find the exact number of D 4 = 2594.
Abstract. Lomonaco and Kauffman introduced knot mosaic system to give a definition of quantum knot system. This definition is intended to represent an actual physical quantum system. A knot (m, n)-mosaic is an m × n matrix of mosaic tiles which are T0 through T10 depicted as below, representing a knot or a link by adjoining properly that is called suitably connected. An interesting question in studying mosaic theory is how many knot (m, n)-mosaics are there. Dm,n denotes the total number of all knot (m, n)-mosaics. This counting is very important because the total number of knot mosaics is indeed the dimension of the Hilbert space of these quantum knot mosaics.In this paper, we find a table of the precise values of Dm,n for 4 ≤ m ≤ n ≤ 6 as below. Mainly we use a partition matrix argument which turns out to be remarkably efficient to count small knot mosaics.
Abstract. The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the trefoil knot 3 1 and the figure-8 knot 4 1 are the only knot types of lattice stick number less than 15, which verifies the result from previous numerical estimations on this quantity.
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