Narrow Results

We give a recurrence formula for calculating the Alexander polynomials of 2-bridge links by using a special type of Conway diagram. As an application we give sufficient and necessary conditions for the periodic covering link over a 2-bridge link to be a fibered link. We also give a recurrence formula to calculate the reduced Alexander polynomials of periodic covering links over 2-bridge links.

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.

For any integers n₁, n₂, …, n_m, we give a formula for the Casson knot invariant of a p-periodic knot (p ≥ 2) whose quotient link is a 2-bridge link with Conway's normal form C(2, 2n₁ + 1, -2, 2n₂ + 1, …, 2n_m + 1, (-1)^m2).

We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper.

We describe the link-symmetric groups of 2-bridge links by Schubert's normal forms and Conway's normal forms. The part of Schubert's normal forms is a collection of well-known facts. However, the part of Conway's normal forms may be new. Moreover, we deal with Conway's normal forms whose entries are even integers.