TY - JOUR

T1 - A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

AU - Kauffman, Louis H.

AU - Ogasa, Eiji

AU - Schneider, Jonathan

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 - Spun-knots (respectively, spinning tori) in S4 made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in S4. 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 submanifolds. The construction proceeds as follows: For a virtual 1-knot K, take an embedded circle C contained in (a closed oriented surface F)×(a closed interval [0, 1]), where F is called a representing surface in virtual 1-knot theory. Embed F in S4 by an embedding map f, and let F stand for f(F). Regard the tubular neighborhood of F in S4 as the result of rotating F × [0, 1] around F. Rotate C together then with F × [0, 1]. When C (F ×{0}) = φ, we obtain an embedded torus Q S4. We prove the following: The embedding type Q in S4 depends only on K, and does not depend on f. Furthermore, the submanifolds, Q and "the embedded torus made from K by using Satoh's method", of S4 are isotopic. Fiberwise equivalence of diagrams refers to fiberwise equivalence of tori in 4-space that lie over the diagrams. 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 (see the body of the paper for this definition). We generalize the construction in the virtual 1-knot case written in the first paragraph, and we also succeed to make a consistent construction of one-dimensional-higher submanifolds from any virtual two-dimensional 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. Hence Rourke's method cannot be generalized to the virtual 2-knot case. Only the spinning construction introduced in this paper works for now.

AB - Spun-knots (respectively, spinning tori) in S4 made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in S4. 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 submanifolds. The construction proceeds as follows: For a virtual 1-knot K, take an embedded circle C contained in (a closed oriented surface F)×(a closed interval [0, 1]), where F is called a representing surface in virtual 1-knot theory. Embed F in S4 by an embedding map f, and let F stand for f(F). Regard the tubular neighborhood of F in S4 as the result of rotating F × [0, 1] around F. Rotate C together then with F × [0, 1]. When C (F ×{0}) = φ, we obtain an embedded torus Q S4. We prove the following: The embedding type Q in S4 depends only on K, and does not depend on f. Furthermore, the submanifolds, Q and "the embedded torus made from K by using Satoh's method", of S4 are isotopic. Fiberwise equivalence of diagrams refers to fiberwise equivalence of tori in 4-space that lie over the diagrams. 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 (see the body of the paper for this definition). We generalize the construction in the virtual 1-knot case written in the first paragraph, and we also succeed to make a consistent construction of one-dimensional-higher submanifolds from any virtual two-dimensional 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. Hence Rourke's method cannot be generalized to the virtual 2-knot case. Only the spinning construction introduced in this paper works for now.

KW - 2-knots

KW - Classical knot theory

KW - Framed welded knots

KW - Ribbon 2-knots

KW - Rotational welded knots

KW - Spun knots

KW - Topological quantum field theory

KW - Virtual 2-knots

KW - Virtual knot theory

KW - Welded knot theory

KW - Welded knots

UR - http://www.scopus.com/inward/record.url?scp=85123946242&partnerID=8YFLogxK

U2 - 10.1142/S0218216521400034

DO - 10.1142/S0218216521400034

M3 - Article

AN - SCOPUS:85123946242

VL - 30

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 10

M1 - 2140003

ER -