Narrow Results

The HC-move was defined as an unknotting operation of a ribbon 2-knot. Representing a ribbon 2-knot by a virtual arc, we see that the HC-move corresponds to one of the "forbidden moves", which unknot every virtual knot. Using this, we consider the α₂-invariant of a ribbon 2-knot and a relation between the Δ-move for a 1-knot and the HC-move for its spun 2-knot.

The transforms of two oriented parallel strands to a k-half twist of two strands are called t_k-move and -move respectively depending on the orientations of the two strands. In this paper we give criterions to detect whether a knot K can be transformed to a knot K' by t_2k-moves and -moves respectively and if so, we give some results on how many moves are needed in these transformations respectively, by using some Vassiliev invariants. Moreover we give a relation between the Δ-move and the t_2k-move by considering the coefficient of z² in the Conway polynomial of a knot, which is a Vassiliev invariant of degree 2.

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, C_n-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by C_n-moves. We define a local move, β_n-move, which is obtained from a C_n-move by duplicating a single pair of arcs with same end points. Then we immediately have that a C_n+1-move is realized by a β_n-move and that a β_n-move is realized by twice C_n-moves. In this note we study the relation between C_n-move and β_n-move, and in particular, give answers to the following questions: (1) Is a β_n-move realized by a finite sequence of C_n+1-moves? (2) Is C_n-move realized by a finite sequence of β_n-moves?

We analyze various local moves on knot diagrams to show that Jones polynomials must have certain algebraic properties. In particular, we show that the Jones polynomial of a knot cannot be a nontrivial monomial.

In 1990, Okada showed that the second coefficients of the Conway polynomials of two knots differ by 1 if the two knots are related by a single Δ-move. We extend the Okada's result for virtual knots by using a Vassiliev invariant v₂ of virtual knots of degree 2 which is induced from the Kauffman polynomial of a virtual knot. We show that v₂(K₁) - v₂(K₂) = ±48, if K₂ is a virtual knot obtained from a virtual knot K₁ by applying a Δ-move. From this we have a lower bound for the number of Δ-moves if two virtual knots K₁ and K₂ are related by a sequence of Δ-moves.