Local moves on knots and products of knots II

Kauffman, Louis H.; Ogasa, Eiji

doi:10.48550/arxiv.1406.5573

Cited by 7 publications

(17 citation statements)

References 12 publications

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“…In [11,12,13,17,19,20,21,22,23], furthermore, we proved several relations between local-moves on n-knots and their invariants (n ∈ N).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local-move-identities for the Z[t,t^{-1}]-Alexander polynomials of 2-links, the alinking number, and high dimensional analogues

Ogasa

2016

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2 ) • ∆L0 holds for three 1links L + , L − , and L 0 which satisfy a famous local-move-relation, where ∆L becomes the Alexander-Conway polynomial of L if we let z = t 1 2 − t −1 2 . We prove a new localmove-identity for the Z[t, t −1 ]-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander-Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the Z[t, t −1 ]-Alexander polynomials of 2-links and the alinking number of 2-links. We show high dimensional analogues of these results. Furthermore we prove that in the 2-dimensional case we cannot normalize the Z[t, t −1 ]-Alexander polynomials to be compatible with our identity but that in a high-dimensional case we can do that to be compatible with our new identity.

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“…In [11,12,13,17,19,20,21,22,23], furthermore, we proved several relations between local-moves on n-knots and their invariants (n ∈ N).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local-move-identities for the Z[t,t^{-1}]-Alexander polynomials of 2-links, the alinking number, and high dimensional analogues

Ogasa

2016

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“…We perturb g| (D 2 1 ∪...∪D 2 n )∩(g −1 (U )) in the special way below but we must remember Note 8. 21. Take an open neighborhood V of U in R 2 b × [0, 1] such that U ⊂ V .…”

Section: The E-equivalencementioning

confidence: 99%

“…See [20,21,22,28,35] for codimension two high dimensional knots. See [6,7,27] for high codimensional high dimensional knots.…”

Section: Virtual High Dimensional Knotsmentioning

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

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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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“…Local moves on high dimensional knots were defined in [19,21,23]. They have been researched in [8,9,10,19,20,21,22,23,24,25]. We show an example of them before we review the definition of high-dimensional pass-moves on high dimensional knots.…”

Section: High-dimensional-pass-moves On High-dimensional Knots and Th...mentioning

confidence: 99%

“…In Proof of Claim 7.3 we also prove that the (2k + 1, 2k + 2)-pass-move-unknottingnumber of P in Proof of Claim 7.3 is≦ 2. (Another proof is given by using Main Theorem 2.6 of [9]. )…”

mentioning

confidence: 99%

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

Ogasa

2016

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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. It is very natural to consider the 'unknotting-number' associated with other local-moves on n-dimensional knots (n ∈ N). In this paper we prove the following facts. For the ribbon-move on 2-knots, which is a kind of local-move on knots, we have the following: There is a ribbon-move-unknotting-number-two 2-knot. The ribbon-move-unknotting-number of 2-knots is unbounded. For the pass-move on 1-knots, which is a kind of local-move on knots, we have the following: There is a pass-move-unknotting-number-two 1-knot whose (ordinary) unknotting-number is 4. For any natural number n, there is a 1-knot whose pass-move-unknotting-number is> n and whose (ordinary) unknotting-number is 4n. For the high-dimensional-pass-move on high-dimensional knots, which is a kind of localmove on knots, we have the following: There is a (2k + 1, 2k + 2)-pass-move-unknottingnumber-two (4k+2)-knot. The (2k+1, 2k+2)-pass-move-unknotting-number of (4k+2)knot is unbounded. There is a (2k+1, 2k+1)-pass-move-unknotting-number-two (4k+1)knot. The (2k + 1, 2k + 1)-pass-move-unknotting-number of (4k + 1)-knot is unbounded. There is a (4k + 1)-knot whose twist-move-unknotting-number is n for any natural number n.

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Local moves on knots and products of knots II

Cited by 7 publications

References 12 publications

Local-move-identities for the Z[t,t^{-1}]-Alexander polynomials of 2-links, the alinking number, and high dimensional analogues

Local-move-identities for the Z[t,t^{-1}]-Alexander polynomials of 2-links, the alinking number, and high dimensional analogues

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

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

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