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Brieskorn submanifolds, local moves on knots, and knot products. / Kauffman, Louis H.; Ogasa, Eiji.
In: Journal of Knot Theory and its Ramifications, Vol. 28, No. 10, 1950068, 01.09.2019.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Brieskorn submanifolds, local moves on knots, and knot products
AU - Kauffman, Louis H.
AU - Ogasa, Eiji
PY - 2019/9/1
Y1 - 2019/9/1
N2 - We first prove the following: Let p ≥ 2 and p ≥. Let K and J be closed, oriented, (2p + 1)-dimensional (p-1)-connected, simple submanifolds of S2p+3. Then K and J are isotopic if and only if a Seifert matrix associated with a simple Seifert hypersurface for K is (-1)p-S-equivalent to that for J. We also discuss the p = 1 case. This result implies one of our main results: Let μ ≥. A 1-link A is pass-equivalent to a 1-link B if and only if A μHopf is (2μ + 1, 2μ + 1)-pass-equivalent to B μHopf. Here, J K means the knot product of J and K, and J μK means JK⋯ Kïμ. See the body of the paper for the definition of knot products. It also implies the other main results: We strengthen the authors' old result that two-fold cyclic suspension commutes with the performance of the twist move for spherical (2k + 1)-knots. See the body for the precise statement. Furthermore, it implies the following: Let p ≥ 2 and p ≥. Let K be a closed oriented (2p + 1)-submanifold of S2p+3. Then K is a Brieskorn submanifold if and only if K is (p-1)-connected, simple and has a (p + 1)-Seifert matrix associated with a simple Seifert hypersurface that is (-1)p-S-equivalent to a KN-type (see the body of the paper for a definition). We also discuss the p = 1 case.
AB - We first prove the following: Let p ≥ 2 and p ≥. Let K and J be closed, oriented, (2p + 1)-dimensional (p-1)-connected, simple submanifolds of S2p+3. Then K and J are isotopic if and only if a Seifert matrix associated with a simple Seifert hypersurface for K is (-1)p-S-equivalent to that for J. We also discuss the p = 1 case. This result implies one of our main results: Let μ ≥. A 1-link A is pass-equivalent to a 1-link B if and only if A μHopf is (2μ + 1, 2μ + 1)-pass-equivalent to B μHopf. Here, J K means the knot product of J and K, and J μK means JK⋯ Kïμ. See the body of the paper for the definition of knot products. It also implies the other main results: We strengthen the authors' old result that two-fold cyclic suspension commutes with the performance of the twist move for spherical (2k + 1)-knots. See the body for the precise statement. Furthermore, it implies the following: Let p ≥ 2 and p ≥. Let K be a closed oriented (2p + 1)-submanifold of S2p+3. Then K is a Brieskorn submanifold if and only if K is (p-1)-connected, simple and has a (p + 1)-Seifert matrix associated with a simple Seifert hypersurface that is (-1)p-S-equivalent to a KN-type (see the body of the paper for a definition). We also discuss the p = 1 case.
KW - Brieskorn manifolds
KW - Brieskorn submanifolds
KW - local moves on 1-knots
KW - local moves on high-dimensional knots, crossing changes on 1-links
KW - pass-moves on 1-links
KW - pass-moves on high-dimensional links
KW - Products of knots
KW - Seifert hypersurfaces
KW - Seifert matrices
KW - twist-moves on high-dimensional links
KW - local moves on high-dimensional knots
KW - crossing changes on 1-links
KW - INVARIANTS
KW - CLASSIFICATION
KW - RIBBON-MOVES
UR - http://www.scopus.com/inward/record.url?scp=85072128048&partnerID=8YFLogxK
U2 - 10.1142/S0218216519500688
DO - 10.1142/S0218216519500688
M3 - Article
AN - SCOPUS:85072128048
VL - 28
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
SN - 0218-2165
IS - 10
M1 - 1950068
ER -
ID: 21541436