Ribbon-move-unknotting-number-two 2-knots, pass-move-unknotting-number-two 1-knots, and high dimensional analogue

Ogasa, Eiji

doi:10.48550/arxiv.1612.03325

Cited by 1 publication

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“…In this paper we discuss them. Local moves on high dimensional knots were defined in [27,29,31] and have been researched in [16,18,27,28,29,30,31,32,33,34,35].…”

Section: Pass-movementioning

confidence: 99%

Local moves on knots and products of knots

Kauffman

Ogasa

2014

Banach Center Publ.

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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 .

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“…In this paper we discuss them. Local moves on high dimensional knots were defined in [27,29,31] and have been researched in [16,18,27,28,29,30,31,32,33,34,35].…”

Section: Pass-movementioning

confidence: 99%