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On the state number of virtual knots

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

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.