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Seminars

On the state number of virtual knots

Hold Date
2015-01-30 16:50〜2015-01-30 17:30
Place
Seminar Room 1, Faculty of Mathematics building, Ito Campus
Object person
 
Speaker
Hitomi KAMIKODANI (Kyusyu University)

Abstract:
Virtual knot theory was proposed by Louis H. Kauffman in 1996 as an extension of the usual knot theory. In knot theory, a knot is a simple closed curve in the 3-space, and is represented by a knot diagram obtained by projecting it to a plane. On the other hand, in virtual knot theory, virtual knot diagrams are defined first, and an equivalence relation among them is defined by using extented Reidemeister moves; then, a virtual knot is defined to be an equivalence class. State numbers, which were defined by T. Nakamura, are invariants of a virtual knot. For a virtual knot diagram, if all of its real crossings are smoothed, then it is called a state, and it is an n-state if it consists of n closed curves. We count the number of n-states for each diagram, and its minimum over all equivalent diagrams is called the n-state number of the virtual knot. State numbers measure the complexity of a virtual knot, while it is in general very difficult to calculate.  For n less than or equal to 3, Nakamura et al. gave some estimates for the n-state numbers; however, no more result has been obtained so far. In this talk, for n greater than or equal to 4, some estimates for the n-state numbers will be given.