An Integral Region Choice Problem on Knot Projection

Abstract

Abstract. In this paper we propose a region choice problem for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.

Lower bounds for the warping degree of a knot projection

“…In this section we explore how to find the independent region sets. A region choice matrix M , defined in [2], of a knot projection P of n crossings is the following n × (n + 2) matrix. (The transposition is known as an incidence matrix defined in [3].)…”

“…1 If it works on Z 2 , it is known that the simultaneous equations have solutions for any b i 's and any region choice matrix of a knot projection ( [7], [3]). Besides, if x i 's are permitted to have the value for any integer, it is also known that the simultaneous equations have solutions for any region choice matrix of a knot projection even if b i 's have the value for any integers ( [2]). In this case, however, the equation has no solutions for some b i 's.…”

“…By taking a base crossing at a crossing which belongs to one of the two bigons, we can take an independent region at the other bigon. For the case that c(P ) ≥ 3, then the number of regions is 3 + 2 = 5 or more by the Euler characteristic (see, for example, [2]). Take a base crossing c. Then either three or four regions are incident to c. This means there exists a region which is not incident to c. Hence IR(P ) ≥ 1 holds.…”

Region crossing change on spatial theta-curves

“…Example 5.2. For the diagram D in Figure 10, if one wants to change the crossing c 1 , one should solve the following simultaneous equations (see [1] for knots): Figure 10: A diagram of a spatial θ-curve.…”

A subspecies of region crossing change, region freeze crossing change

“…Region crossing change is an unknotting operation for knots, because any link is untied by a sequence of crossing changes. Varieties of region crossing change have been proposed and studied by several authors [1,3]. In this paper, we introduce another subspecies of region crossing change named region freeze crossing change.…”