Studies on geometric properties and realization problems of invariants for classical knots and virtual knots

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In this studies, we obtain several results about realizations for geometric or algebraic properties of knots and related topics from a view of local moves for knots. In fact, we construct p-periodic knots with C_n unknotting number one for any period p. We also construct composite knots with # unknotting number one. Moreover, we obtain some partial answers for a realization problem of Alexander polynomials for # unknotting number one knots. On the other hand, we construct non-classical virtual knots with the trivial Miyazawa polynomial, which is a generalization of Jones polynomial.