Minimal hard surface-unlink and classical unlink diagrams

Abstract

Dedicated to Witold Rosicki on his 65th birthday.Abstract. We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime surface-unknot and surface-unlink diagrams with prime base surface components up to ten crossings.

Triple-crossing projections, moves on knots and links and their minimal diagrams

“…The moves J R and J R are independent of each another, because J R (as oppose to J R ) move changes the number of triple-crossings of name eY (in an sPD code of the diagram) shown in Figure 19, therefore we have the following. We denote this set as Sh n , and generate every element of it just as in one of the authors earlier papers (in [5,Section 3.]). The enumeration of elements of this set (for an even n) is presented in Table 1 (second column), we associate each such projection with its PD code (described in that paper and in [4]).…”

“…Then, we re-numerate each sPD code with consecutive positive integers from one up to three times the number of triple-crossings, with increasing labels as we go around each transverse circular component in the tripleprojection. An example is presented in Figure 13, where a knot diagram with code PD[X [1,4,2,5],X [3,8,4,9],X [12,6,13,5],X [13,16,14,1],X[9,14,10,15],X [15,10,16,11],X [6,12,7,11],X [7,2,8,3]] is transform, by contracting four edges labeled: 5,8,11,14, to a three-component link diagram with code sPD[eX [1,4,2,12,6,13],eX [4,9,3,3,7,2],eX [15,10,16,6,12,7],eX [1,13,...…”

“…For our algorithm, we use the following (well-known) general method for testing spherical graphs isotopy (see [8]). 5 τ 1 = (1,7,10,8,12,11) (2,4,5,9,3,6) σ 1 =(1,5,10)(2,12)(4,8) (6,9,11) σ 2 =(1,10,12)(2,6,11)(3,7)(5,9) 1,3,5,6,10,4) (2,8,11,9,7,12) Any isotopy class of connected graphs with e number of edges embedded in the sphere corresponds to a pair of permutations σ, τ on 2e elements, that σ · τ has order two, and σ, τ are determined up to simultaneous conjugacy. The correspondence is as follows.…”

“…Is there a prime knot K with an odd number c 2 (K) = 3c 3 (K)? [4,2,5,5,1,6], eY [3,1,2,4,6,3]] 10 136 sPD [eX[9, 2, 10, 10, 1, 11], eX [6,4,7,7,3,8], eY [4, 3, 12, 5, 2, 1], eY[11, 9, 5, 12, 8, 6]] 10 137 sPD [eX[9, 2, 10, 10, 1, 11], eY [7,4,6,8,3,7], eY [5,12,8,6,11,9], eY [2,1,4,3,12,5]] 10 140 sPD[eY [4,10,1,5,9,2], eY [2,12,6,3,11,7], eY [12,9,5,...…”

“…K12n 462 sPD[eX [4,2,9,5,1,10], eY [10,4,7,11,3,8], eY [6,3,11,7,2,12], eX [8, 1, 5, 9, 12, 6]] [1,1,3,2,2,3]] [4,3,5,5,2,6], eY [3,4,1,1,6,2]] 6 3 3 sPD[eX [1,4,6,2,3,5], eX [5,3,2,6,4,1]] 5 2 1 sPD[eX [8,1,7,9,2,8], eY <...>…”