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CASSON KNOT INVARIANTS OF PERIODIC KNOTS WITH RATIONAL QUOTIENTS

HEE JEONG JANG

Department of Mathematics, Graduate School of Natural Sciences, Pusan National University, Pusan 609-735, Korea

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SANG YOUL LEE

Department of Mathematics, Pusan National University, Pusan 609-735, Korea

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, and

MYOUNGSOO SEO

Department of Mathematics, Pusan National University, Pusan 609-735, Korea

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https://doi.org/10.1142/S0218216507005373Cited by:8 (Source: Crossref)

Abstract

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.

Keywords:

AMSC: 57M25

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