Make your Boy surface

Ogasa, Eiji

doi:10.48550/arxiv.1303.6448

Cited by 2 publications

(2 citation statements)

References 4 publications

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“…We show an example. Take the Boy surface in R 3 (see [3,34]). We can regard it as a virtual 2-knot diagram as follows: Suppose that the only one immersed crossing curve is a virtual one.…”

Section: The Virtual 2-knot Casementioning

confidence: 99%

A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

Kauffman¹,

Ogasa²,

Schneider³

2018

Preprint

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Spun-knots (respectively, spinning tori) in S 4 made from classical 1-knots compose an important class of spherical 2-knots (respectively, embedded tori) contained in S 4 . Virtual 1-knots are generalizations of classical 1-knots. We generalize these constructions to the virtual 1-knot case by using what we call, in this paper, the spinning construction of a submanifold. The construction proceeds as follows: It has been known that there is a consistent way to make an embedded circle C contained in (a closed oriented surface F )×(a closed interval [0, 1]) from any virtual 1-knot K. Embed F in S 4 by an embedding map f . Let F also denote f (F ). We can regard the tubular neighborhood ofThus we obtain an embedded torus Q ⊂ S 4 . We prove the following: The embedding type Q in S 4 depends only on K, and does not depend on f . Furthermore, the submanifolds, Q and the embedded torus made from K, defined by Satoh's method, of S 4 are isotopic.We generalize this construction in the virtual 1-knot case, and we also succeed to make a consistent construction of one-dimensional-higher tubes from any virtual 2dimensional knot. Note that Satoh's method says nothing about the virtual 2-knot case. Rourke's interpretation of Satoh's method is that one puts 'fiber-circles' on each point of each virtual 1-knot diagram. If there is no virtual branch point in a virtual 2-knot diagram, our way gives such fiber-circles to each point of the virtual 2-knot diagram. Furthermore we prove the following: If a virtual 2-knot diagram α has a virtual branch point, α cannot be covered by such fiber-circles.We obtain a new equivalence relation, the E-equivalence relation of the set of virtual 2-knot diagrams, that is much connected with the welded equivalence relation and our spinning construction. We prove that there are virtual 2-knot diagrams, J and K, that are virtually nonequivalent but are E-equivalent.Although Rourke claimed that two virtual 1-knot diagrams α and β are fiberwise equivalent if and only if α and β are welded equivalent, we state that this claim is wrong. We prove that two virtual 1-knot diagrams α and β are fiberwise equivalent if and only if α and β are rotational welded equivalent (the definiton of rotational welded equivalence is given in the body of the paper).

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“…We show an example. Take the Boy surface in R 3 (see [3,34]). We can regard it as a virtual 2-knot diagram as follows: Suppose that the only one immersed crossing curve is a virtual one.…”

Section: The Virtual 2-knot Casementioning

confidence: 99%

A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

Kauffman¹,

Ogasa²,

Schneider³

2018

Preprint

Self Cite

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“…We use Boy surface. See In [44] the author have created a way to construct the Boy surface by using a pair of scissors, a piece of paper, and a strip of scotch tape.…”

Section: The Projections Of N-dimensional Knotsmentioning

confidence: 99%

Introduction to high dimensional knots

Ogasa

2013

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This is an introductory article on high dimensional knots for the beginners. High dimensional knot theory is an exciting field. It is a field of knot theory, which is one of topology and is connected with many ones.In this article we use few literal expressions, equations, functions, etc. We barely suppose that the readers have studied manifolds, homology theory, or topics beyond them.Is there a truly knotted high dimensional knot? We first answer this question. After that, we explain local moves on high dimensional knots and the projections of high dimensional knots.

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Make your Boy surface

Cited by 2 publications

References 4 publications

A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

Introduction to high dimensional knots

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