Narrow Results

It is shown by Y. Ohyama, K. Taniyama and S. Yamada that for any natural number n and any knot K, there are infinitely many unknotting number one knots, whose Vassiliev invariants of order less than or equal to n coincide with those of K. We analyze it for delta unknotting number, and obtain the following. For any natural number n and any oriented knots K and M with a₂(K)≠a₂(M) there are infinitely many knots J_m such that the delta distance between J_m and M coincide with |a₂(K)-a₂(M)| and whose Vassiliev invariants of order less than or equal to n coincide with those of K. Here a₂(K) is the second coefficient of the Conway polynomial of K.

Two spatial embeddings of a graph are said to be edge-homotopic if they are transformed into each other by self-crossing changes and ambient isotopies. We show that two spatial embeddings of the complete graph on four vertices are edge-homotopic if and only if they have the same α-invariant.

We consider the metric space of all knots on which the distance is defined by delta moves. We show that for any two knots K₁ and K₂ with delta distance k and for any natural numbers ℓ and m with ℓ + m = k, the intersection of the ball of radius ℓ centered at K₁ and the ball of radius m centered at K₂ contains infinitely many knots. We also consider the problem whether or not the center of a given ball is unique.

For any knot K represented as an n-braid (n ≥ 3), we construct an infinite sequence of pairwise non-conjugate (n + 1)-braids {b_m, m ∈ ℕ} representing K.

For classical knots, $\mathrm{\Delta }$-moves and crossing changes can both unknot any knot. But many virtual knots cannot be unknotted with these moves. Moreover, there are many virtual knots that can be unknotted by crossing changes but cannot be unknotted by $\mathrm{\Delta }$-moves. We show that the derivatives of the Jones polynomial can detect whether a virtual knot can be unknotted by $\mathrm{\Delta }$-moves.