The Intersection of Spheres in a Sphere and a New Geometric  Meaning of the Arf Invariant

Ogasa, Eiji

doi:10.1142/s0218216502002104

Cited by 8 publications

(14 citation statements)

References 16 publications

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“…Regarding the pairing as the sum of the linking numbers of "0-dimensional Hopf links S 0 ⊔ S 0 ֒→ R 1 " determined by the crossings of the diagrams, we can say that Theorem 2.5 is a higher-dimensional analogue to (2.3) and to the Polyak-Viro formula. Theorem 2.5 together with the result of Ogasa [21] gives an alternative proof for the following result of Murai-Ohba [22], which states that the "unknotting number" of any nontrivial embedding f ∈ K 6,3 is 1 (see §6.4 for the proof). Corollary 2.7 ([22]).…”

Section: Notations and Resultsmentioning

confidence: 99%

“…The double point set of generic g ∈ I 5,3 is a classical link. A result of Ogasa [21] characterizes which link can be realized as a double point set of an immersion R 3 R 5 . Theorem 6.10 (A special case of [21, Theorem 1.1]).…”

Section: Casementioning

confidence: 99%

“…Theorem 2.5 together with the result of Ogasa [21] gives an alternative proof for the following result of Murai-Ohba [22], which states that the "unknotting number" of any nontrivial embedding f ∈ K 6,3 is 1 (see §6.4 for the proof).…”

Section: Definition 24 (See Figure 21) We Say An Embeddingmentioning

confidence: 99%

“…Namely, f is isotopic to some f ′ , with f ′ {1} isotopic to the trivial inclusion. The proof is outlined as follows: Given any two-component link L ⊂ R 3 , by a result of Ogasa [21] we can find f 0 : R 3 ֒→ R 6 such that it is isotopic to the trivial inclusion and its projection p • f : R 3 R 5 has L as its double point set. By Theorem 2.5 the embedding f 1 obtained by the crossing change along L satisfies H( f 1 ) = −lk(L).…”

Section: Definition 24 (See Figure 21) We Say An Embeddingmentioning

confidence: 99%

“…Contrary to the expectation, we see in Theorem 2.11 that in general E is not of order 1. In fact, we see that, when k = 1, the jump of E at self-tangency singularities can be computed using the linking numbers of links that arise as the double point set of immersion R 3 R 5 , and the linking numbers can be arbitrarily large by the result of Ogasa [21]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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Lin–Wang type formula for the Haefliger invariant

Sakai

2015

Homology, Homotopy and Applications

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In this paper we study the Haefliger invariant for long embeddings R 4k−1 ֒→ R 6k in terms of the self-intersections of their projections to R 6k−1 , under the condition that the projection is a generic long immersion R 4k−1 R 6k−1 . We define the notion of "crossing changes" of the embeddings at the self-intersections and describe the change of the isotopy classes under crossing changes using the linking numbers of the double point sets in R 4k−1 . This formula is a higher-dimensional analogue to that of X.-S. Lin and Z. Wang for the order 2 invariant for classical knots. As a consequence, we show that the Haefliger invariant is of order two in a similar sense to Birman and Lin. We also give an alternative proof for the result of M. Murai and K. Ohba concerning "unknotting numbers" of embeddings R 3 ֒→ R 6 . Our formula enables us to define an invariant for generic long immersions R 4k−1 R 6k−1 which are liftable to embeddings R 4k−1 ֒→ R 6k . This invariant corresponds to V. Arnold's plane curve invariant in Lin-Wang theory, but in general our invariant does not coincide with order 1 invariant of T. Ekholm.Date: October 13, 2018. Key words and phrases. the space of embeddings, the Haefliger invariant, configuration space integral, finite type invariant, generic immersion.

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Section: Notations and Resultsmentioning

confidence: 99%

Section: Casementioning

confidence: 99%

Section: Definition 24 (See Figure 21) We Say An Embeddingmentioning

confidence: 99%

Section: Definition 24 (See Figure 21) We Say An Embeddingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lin–Wang type formula for the Haefliger invariant

Sakai

2015

Homology, Homotopy and Applications

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The “unknotting number” associated with other local moves than the crossing-change

Ogasa

2018

J. Knot Theory Ramifications

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The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let [Formula: see text] be a positive integer. It is very natural to consider the “unknotting number” associated with other local moves on [Formula: see text]-dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The “unknotting number” associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is [Formula: see text]. For any positive integer [Formula: see text], there is a 1-knot whose “unknotting number” associated with the pass-move is [Formula: see text] and whose ordinary unknotting number is [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers. For the [Formula: see text]-move on [Formula: see text]-knots, which is a local move on knots, we have the following: Let [Formula: see text] be a non-negative integer. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one time. The “unknotting number” associated with the [Formula: see text]-move is unbounded. There is a [Formula: see text]-knot which is changed into the unknot by two times of the [Formula: see text]-move not by one. The “unknotting number” associated with the [Formula: see text]-move is unbounded. We prove the following: For any positive integer [Formula: see text] and any positive integer [Formula: see text], there is a [Formula: see text]-knot which is changed into the unknot by [Formula: see text] times of the twist-move not by [Formula: see text] times.

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Brieskorn submanifolds, local moves on knots, and knot products

Kauffman

Ogasa

2019

J. Knot Theory Ramifications

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We first prove the following: Let p ≥ 2 and p ∈ N. Let K and J be closed, oriented, (2p + 1)-dimensional connected, (p − 1)-connected, simple submanifolds of S 2p+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:It also implies the other of our 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 ∈ N. Let K be a closed oriented (2p + 1)-submanifold of S 2p+3 . Then K is a Brieskorn submanifold if and only if K is connected, (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.(2) Let K and J be closed oriented 3-dimensional simple submanifolds of S 5 . Then the following two statements are equivalent. (i) K is isotopic to J.(ii) There is a simple Seifert hypersurface V K (respectively, V J ) for K (respectively, J) with the following properties: There is an orientation preserving diffeomorphism map f : V K → V J . V K (respectively, V J

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The Intersection of Spheres in a Sphere and a New Geometric Meaning of the Arf Invariant

Cited by 8 publications

References 16 publications

Lin–Wang type formula for the Haefliger invariant

Lin–Wang type formula for the Haefliger invariant

The “unknotting number” associated with other local moves than the crossing-change

Brieskorn submanifolds, local moves on knots, and knot products

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