TY - JOUR

T1 - Local-moves on knots and products of knots II

AU - Kauffman, Louis H.

AU - Ogasa, Eiji

N1 - Kauffman's work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (Contract No. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).
Publisher Copyright:
© 2021 World Scientific Publishing Company.

PY - 2021/9/1

Y1 - 2021/9/1

N2 - We use the terms, knot product and local-move, as defined in the text of this paper. Let n be an integer 3. Let n be the set of simple spherical n-knots in Sn+2. Let m be an integer 4. We prove that the map j: 2m →2m+4 is bijective, where j(K) = KS - Hopf, and Hopf denotes the Hopf link. Let J and K be 1-links in S3. Suppose that J is obtained from K by a single pass-move, which is a local-move on 1-links. Let k be a positive integer. Let P S - QS - k denote the knot product P S - Q S - ⋯ S - Qk. We prove the following: The (4k + 1)-dimensional submanifold JS - HopfS - k S S4k+3 is obtained from KS - HopfS - k by a single (2k+1, 2k+1)-pass-move, which is a local-move on (4k + 1)-submanifolds contained in S4k+3. See the body of this 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 manifolds, ς(a,b, 2, 22k) and ς(a',b', 2, 2 2k), are diffeomorphic as abstract manifolds. Let J and K be (not necessarily connected or spherical) 2-dimensional closed oriented submanifolds in S4. Suppose that J is obtained from K by a single ribbon-move, which is a local-move on 2-dimensional submanifolds contained in S4. Let k be an integer ≥ 2. We prove the following: The (4k + 2)-submanifold JS - HopfS - k S S4k+4 is obtained from KS - HopfS - k by a single (2k + 1, 2k + 2)-pass-move, which is a local-move on (4k + 2)-dimensional submanifolds contained in S4k+4.

AB - We use the terms, knot product and local-move, as defined in the text of this paper. Let n be an integer 3. Let n be the set of simple spherical n-knots in Sn+2. Let m be an integer 4. We prove that the map j: 2m →2m+4 is bijective, where j(K) = KS - Hopf, and Hopf denotes the Hopf link. Let J and K be 1-links in S3. Suppose that J is obtained from K by a single pass-move, which is a local-move on 1-links. Let k be a positive integer. Let P S - QS - k denote the knot product P S - Q S - ⋯ S - Qk. We prove the following: The (4k + 1)-dimensional submanifold JS - HopfS - k S S4k+3 is obtained from KS - HopfS - k by a single (2k+1, 2k+1)-pass-move, which is a local-move on (4k + 1)-submanifolds contained in S4k+3. See the body of this 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 manifolds, ς(a,b, 2, 22k) and ς(a',b', 2, 2 2k), are diffeomorphic as abstract manifolds. Let J and K be (not necessarily connected or spherical) 2-dimensional closed oriented submanifolds in S4. Suppose that J is obtained from K by a single ribbon-move, which is a local-move on 2-dimensional submanifolds contained in S4. Let k be an integer ≥ 2. We prove the following: The (4k + 2)-submanifold JS - HopfS - k S S4k+4 is obtained from KS - HopfS - k by a single (2k + 1, 2k + 2)-pass-move, which is a local-move on (4k + 2)-dimensional submanifolds contained in S4k+4.

KW - branched cyclic covering spaces

KW - Local-moves on 1-knots

KW - local-moves on high-dimensional knots

KW - products of knots

KW - q) -pass-move on high-dimensional links

KW - Seifert hypersurfaces

KW - Seifert matrices

KW - the (p

KW - the crossing-change on 1-links

KW - the pass-move on 1-links

KW - the (p, q) -pass-move on high-dimensional links

UR - http://www.scopus.com/inward/record.url?scp=85122311182&partnerID=8YFLogxK

U2 - 10.1142/S021821652140006X

DO - 10.1142/S021821652140006X

M3 - Article

AN - SCOPUS:85122311182

VL - 30

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 10

M1 - 2140006

ER -