Multiplication of Classical Knots
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MULTIPLICATION OF CLASSICAL KNOTS V. M. Nezhinskij∗ and V. V. Nesterenok†
UDC 515.162.8
For oriented one-dimensional knots in three-dimensional Euclidean space, an operation similar to the commutator multiplication of loops is defined, and its simplest properties are studied. Bibliography: 2 titles.
1. Preliminaries. A link is an oriented smooth compact one-dimensional submanifold of R3 such that each of its connected components is diffeomorphic to a circle. If a link consists of a single component, then it is also called a knot. In particular, the circle of unit radius centered at (0, 0, 0) ∈ R3 , located in the plane R2 × 0 (= R × R × 0) ⊂ R3 (= R × R × R), and standardly oriented is a knot. We will call this knot a standard trivial one. An isotopy of R3 is a smooth map F : R3 × I → R3 , where I = [0, 1], such that the restriction of F to R3 × t is a diffeomorphism for any t ∈ I and F (x, 0) = x for any x ∈ R3 . Two links L0 and L1 are said to be isotopic if there is an isotopy F of R3 such that F (L0 × 1) = L1 and F maps each component of L0 × 1 to the corresponding component of L1 preserving orientation. It is well known and easy to prove that the isotopy of links is an equivalence relation. 2. The set K and the map [ , ]. We decompose the set of knots into isotopy classes and denote the resulting set of classes by K. We define the map [ , ]: K × K → K in the following way. Let α, β ∈ K. In the class α, we choose a knot contained in the half-space R3− = {(x, y, z) ∈ R3 |x ≤ 0} such that the intersection of this knot with the plane 0 × R × R (⊂ R3 ) is the segment 0 × [−1, 1] × 0 with the standard orientation. We denote this knot by K1 . In the class β, we choose a knot contained in the half-space R3+ = {(x, y, z) ∈ R3 |x ≥ 0} such that the intersection of this knot with the plane 0 × R × R is the segment 0 × 0 × [−1, 1] with the standard orientation. We denote this knot by K2 . (Obviously, such knots in the classes α and β do exist.) It is not difficult to see that the following two assertions are true. (a) There exists a smooth oriented surface V1 in R3− such that • the pair (V1 , K1 ) is diffeomorphic to the pair (S 1 × [−1, 1], S 1 × 0), • some (and therefore any) connected component of the boundary ∂V1 has zero linking number with K1 , • V1 ∩ (0 × R × R) = 0 × [−1, 1] × [−1, 1], • (∂V1 ) ∩ (0 × R × R) = 0 × [−1, 1] × ({−1} ∪ {1}), ∗
St.Petersburg State University and Herzen State Pedagogical University of Russia, St.Petersburg, Russia, e-mail: [email protected]. †
Herzen State Pedagogical University of Russia, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 476, 2018, pp. 134–142. Original article submitted December 7, 2018. 518 1072-3374/20/2514-0518 ©2020 Springer Science+Business Media, LLC
Fig. 1 • the orientation of V1 induces the standard orientation of the square 0× [−1, 1]× [−1, 1]. (b) There exists a smooth oriented surface V2 in R3+ such that
• the pair (V2 , K2 ) is diffeomorphic to the pair (S 1 × [−1, 1], S 1 × 0), • some (and therefore
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