Local moves on knots and products of knots

Abstract

We use the terms, knot product and local move, as defined in the text of the paper. Let n be an integer≧ 3. Let S n be the set of simple spherical n-knots in S n+2 . Let m be an integer≧ 4. We prove that the map j : S 2m → S 2m+4 is bijective, where j(K) = K⊗Hopf, and Hopf denotes the Hopf link.Let J and K be 1-links in S 3 . Suppose that J is obtained from K by a single passmove, which is a local-move on 1-links. Let k be a positive integer. Let P ⊗ k Q denote the knot product P ⊗ Q ⊗ ... ⊗ Q k . We prove the following: The (4k + 1)-dimensional submanifold J ⊗ k Hopf ⊂ S 4k+3 is obtained from K ⊗ k Hopf by a single (2k + 1, 2k + 1)pass-move, which is a local-move on (4k + 1)-submanifolds contained in S 4k+3 . See the body of the paper for the definitions of all local moves in this abstract.We prove the following: Let a, b, a ′ , b ′ and k be positive integers. If the (a, b) torus link is pass-move equivalent to the (a ′ , b ′ ) torus link, then the Brieskorn manifold Σ(a, b, 2, ..., 2 2k ) is diffeomorphic to Σ(a ′ , b ′ , 2, ..., 2 2k ) as abstract manifolds.Let J and K be (not necessarily connected or spherical) 2-dimensional closed oriented submanifolds in S 4 . Suppose that J is obtained from K by a single ribbon-move, which is a local-move on 2-dimensional submanifolds contained in S 4 . Let k be an integer≥ 2. We prove the following: The (4k + 2)-submanifold J ⊗ k Hopf ⊂ S 4k+4 is obtained from K ⊗ k Hopf by a single (2k + 1, 2k + 2)-pass-move, which is a local-move on (4k + 2)dimensional submanifolds contained in S 4k+4 .

Brieskorn submanifolds, local moves on knots, and knot products

“…These theorems allow us to make comparisons of the action of the knot product (such as taking a knot product with a Hopf link) and certain local moves on high dimensional knots. In Theorem 4.2, we strengthen the authors' old results in [15,16] that taking a knot product with the Hopf link commutes with the performance of the pass move. In Theorem 4.3, we strengthen the authors' old results in [15,16] that two-fold cyclic suspension commutes with the performance of the twist move.…”

“…In Theorem 4.2, we strengthen the authors' old results in [15,16] that taking a knot product with the Hopf link commutes with the performance of the pass move. In Theorem 4.3, we strengthen the authors' old results in [15,16] that two-fold cyclic suspension commutes with the performance of the twist move. Much of the work in this paper depends upon familiarity with knot products and local moves on knots.…”

“…In [15,16] we proved the following: For any simple n-knot K in S n+2 (n ∈ N), take K⊗Hopf. Then we obtain a map S n → S n+4 , where S n is a set of simple n-knots.…”

“…In [15] we began to research relations between knot products and local-moves on knots. This paper is a sequel to [15,16]. Knot products, or products of knots, were defined and have been researched in [11,14,15].…”

“…In more recent work ( [15,16]), the authors of the present paper have worked with other constructions (local moves on n-knots for any natural number n, and generalized band-passing) in relation to the knot product construction. In all these cases it has been important to study the Seifert pairing for manifolds bounding the knots that we study.…”