arXiv subjects
Publications and source records attributed to Kazumichi Nakamura.
The $Δ$-unknotting number for a knot is defined as the minimum number of $Δ$-moves needed to deform the knot into the trivial knot. In this paper, we discuss Montesinos knots whose $Δ$-unknotting number is equal to one. We propose a conjectural characterization of Montesinos knots with $Δ$-unknotting number one and provide examples admitting distinct $Δ$-moves, each of which deforms the knot into the trivial knot.
The $Δ$-unknotting number for a knot is defined as the minimum number of $Δ$-moves needed to deform the knot into the trivial knot. In this paper, we determine the $Δ$-unknotting numbers for certain families of Montesinos knots. Using results on pretzel knots and two-bridge knots, we prove that for these families the $Δ$-unknotting number equals the absolute value of the second coefficient of the Conway polynomial. In particular, every positive pretzel knot belongs to these families, and certain two-bridge knots also belong to them.
The $Δ$-unknotting number for a knot is defined as the minimum number of $Δ$-moves needed to deform the knot into the trivial knot. It is known that, for positive pretzel knots, the $Δ$-unknotting number coincides with the second coefficient of their Conway polynomial. In this paper, we compute the $Δ$-unknotting number for positive pretzel knots. As a consequence of the above result, among positive pretzel knots of odd type with a fixed crossing number $n$, where $n$ is odd, the $Δ$-unknotting number is maximized by $P(1, 1, ... , 1) \quad \big( \cong T(2,n) \big)$, and the maximum value is $\frac{1}{8}(n^2 - 1)$. We also obtain a similar result for torus knots. We further determine the $Δ$-unknotting number for pretzel knots of type $P(-1, p_2, ..., p_n)$, where $p_i$ is a positive odd integer for $2 \leq i \leq n$ and $n$ is odd.
The $Δ$-unknotting number for a knot is defined as the minimum number of $Δ$-moves needed to deform the knot into the trivial knot. We determine the $Δ$-unknotting numbers for two-bridge knots of type $C(2β_1, 2β_2, ... , 2β_n)$ and type $C(2β_1, 2β_2, ... , 2β_{n-1}, 2β_n-1)$, where $β_i$ is a positive integer for $1 \leq i \leq n$. We also discuss two-bridge knots whose $Δ$-unknotting number is equal to one.