n-dimensional links, their components, and their band-sums
Abstract
We prove the following results (1) (2) (3) on relations between n-links and their components. (1) Let L=(L1, L2) be a (4k+1)-link (4k+1≥ 5). Then we have Arf L=Arf L1+Arf L2. (2) Let L=(L1, L2) be a (4k+3)-link (4k+3≥3). Then we have σ L=σ L1+σ L2. (3) Let n≥1. Then there is a nonribbon n-link L=(L1, L2) such that Li is a trivial knot. We prove the following results (4) (5) (6) (7) on band-sums of n-links. (4) Let L=(L1, L2) be a (4k+1)-link (4k+1≥ 5). Let K be a band-sum of L. Then we have Arf K=Arf L1+Arf L2. (5) Let L=(L1, L2) be a (4k+3)-link (4k+3≥3). Let K be a band-sum of L. Then we have σ K=σ L1+ σ L2. The above (4)(5) imply the following (6). (6) Let 2m+1≥3. There is a set of three (2m+1)-knots K0, K1, K2 with the following property: K0 is not any band-sum of any n-link L=(L1, L2) such that Li is equivalent to Ki (i=1,2). (7) Let n≥1. Then there is an n-link L=(L1, L2) such that Li is a trivial knot (i=1,2) and that a band-sum of L is a nonribbon knot. We prove a 1-dimensional version of (1). (8) Let L=(L1, L2) be a proper 1-link. Then Arf L =Arf L1+ Arf L2+1/2\β*(L)+mod4 \1/2lk (L)\\ =Arf L1+Arf L2+mod2 \λ (L)\, where β*(L) is the Saito-Sato-Levine invariant and λ(L) is the Kirk-Livingston invariant.
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