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Classification of genus 1 virtual knots having at most five classical crossings
Preview AbstractThe goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.
Adjoint twisted Alexander polynomials of genus one two-bridge knots
Preview AbstractWe give explicit formulas for the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of genus one two-bridge knots.