Narrow Results

We give a formula for the Casson knot invariant of a p-periodic knot in S³ whose quotient link is a 2-bridge link with Conway's normal form C(2, 2n₁, -2, 2n₂, …, 2n_2m, 2) via the integers p, n₁, n₂, …, n_2m(p ≥ 2 and m ≥ 1). As an application, for any integers n₁, n₂, ≥, n_2m with the same sign, we determine the Δ-unknotting number of a p-periodic knot in S³ whose quotient is a 2-bridge link C(2, 2n₁, -2, 2n₂, ≥, 2n_2m, 2) in terms of p, n₁, n₂, ≥, n_2m. In addition, a recurrence formula for calculating the Alexander polynomial of the 2-bridge knot with Conway's normal form C(2n₁, 2n₂, ≥, 2n_m) via the integers n₁,n₂, ≥, n_m is included.

In this paper, we give a bound for the Δ-unknotting number of a Whitehead double in terms of the unknotting number and a certain integral invariant of its companion knot. As applications, we show that the Δ-unknotting number of m-twisted Whitehead doubles of certain knots does not remember its companion knot, and is equal to the twist number m. We also give possible Δ-unknotting number of m-twisted Whitehead doubles whose companions are knots with unknotting number 1, certain twist knots, amphicheiral knots, and positive knots.

The Δ-unknotting number for a knot is defined to be the minimum number of Δ-unknotting operations which deform the knot into the trivial knot. We determine the Δ-unknotting numbers for torus knots, positive pretzel knots, and positive closed 3-braids.

Delta-unknotting operation is a local move, as shown in Figure 2. Murakami-Nakanishi (MN) showed this move to be an unknotting operation. We will show that for many periodic knots the Δ-unknotting number is greater than one.