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Kauffman-Harary conjecture for alternating virtual knots. (English) Zbl 1335.57012
A knot is $$p$$-colorable for a prime $$p$$ if there is a coloring of the arcs in a diagram of the knot with integers mod $$p$$ such that the colors on the undercrossing arcs at every crossing sum to twice the color on the overcrossing arc mod $$p$$ and at least two distinct colors are used; this is equivalent to the existence of a surjective homomorphism from the fundamental kei of the knot to the Takasaki kei (also known as dihedral quandle or cyclic quandle) structure on the integers mod $$p$$. Not every diagram with such a coloring necessarily uses all of the colors, and one can define the minimal coloring number of a knot mod $$p$$ to be the minimal number of colors used in a nontrivial $$p$$-coloring of a knot $$K$$ over the set of all diagrams equivalent to $$K$$. A coloring is heterogenous if every arc has a different color, i.e. if the minimal coloring number equals the crossing number. In this paper, it is shown that alternating virtual knots with prime determinant $$p$$ and without nugatory classical crossings have the Kauffman-Harary property, i.e., that nontrivial $$p$$-colorings of such knots are always heterogeneous.
##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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