High – Dimensional Knotting Illustrated Guide – Dennis Roseman

6.129 The circles M∗ θ . Fig. 6.130 Projection of Klein Bottle Example 6.25. We define mθ to be the half-flip isotopy in R3 + of the circle M0 ⊆R3 + parameterized by 0 ≤θ ≤2π where mθ is the circle M0 rotated positively about the x-axis by an angle of θ/2. These circles are called meridians of the Klein bottle. This rotation will be considered positive if, looking down the x-axis, it appears to be counterclockwise.

As a set m0 = m2π. We define Mθ = iθ(mθ) ⊆H3 θ. We take the standard knotting of the Klein bottle to be: K2 = Sθ=2π θ=0 Mθ. Figure 6.129 shows the projection of a number of the meridians of K2. For each θ, Mθ ⊆H3 θ ⊆R4 and M ∗ θ ⊆H2 θ ⊆R3. Circle M ∗ 0 projects to a circle. The other circles Mθ project to ellipses with M ∗ π projecting to a line segment.

The resulting projection K2 ∗is shown in Figure 6.130. The crossing set of K2 ∗is a line segment, M ∗ π with each endpoint a branch point. Figures 6.131 and 6.132 give more detail for the w-coordinate, using wire-frame coding. Fig. 6.131 Projection of Klein Bottle with wire-frame coding. Fig. 6.132 Figure 6.131 from a lower viewpoint. We used a rotation of angle θ/2 for 0 ≤θ ≤2π but if we rotate by an angle of −θ/2 (that is a negative rather than positive half-flip), we get a different isotopy, m′ θ and a corresponding different knotting K′.

We will show in Section 7.3 that these are isotopic. High-Dimensional Knotting: An Illustrated Guide Remark 6.4. In topological terms one can say: “the Klein bottle is ob- tained as a connected sum of two proper M¨obius bands”. We can visualize this using K2 ∗as follows. Express the circle mθ is the union of two semi-circles with common endpoints: A = {(r, θ, z, w) ∈mθ : 2 ≤r ≤3} and B = {(r, θ, z, w) ∈mθ : 1 ≤r ≤3}, see Figure 6.133.

Editor-in-charge: Louis H. Kauffman (Univ. of Illinois, Chicago) The Series on Knots and Everything: is a book series polarized around the theory of knots. Volume 1 in the series is Louis H Kauffman’s Knots and Physics. One purpose of this series is to continue the exploration of many of the themes indicated in Volume 1. These themes reach out beyond knot theory into physics, mathematics, logic, linguistics, philosophy, biology and practical experience. All of these outreaches have relations with knot theory when knot theory is regarded as a pivot or meeting place for apparently separate ideas.

Knots act as such a pivotal place. We do not fully understand why this is so. The series represents stages in the exploration of this nexus. Details of the titles in this series to date give a picture of the enterprise. Published: Vol. 79: High-Dimensional Knotting: An Illustrated Guide by D. Roseman Vol. 78: Four-Dimensional Paper Constructions After Möbius, Klein and Boy by E. Ogasa Vol. 77: Quipu: Decorated Permutation Representations of Finite Groups by Y.

Bae, J. S. Carter & B. Kim Vol. 76: Combinatorial Knot Theory by R. A. Fenn Vol. 75: Scientific Legacy of Professor Zbigniew Oziewicz: Selected Papers from the International Conference “Applied Category Theory Graph-Operad-Logic” edited by H. M. C. García, José de Jesús Cruz Guzmán, L. H. Kauffman & H. Makaruk Vol. 74: Seeing Four-Dimensional Space and Beyond: Using Knots!

by E. Ogasa Vol. 73: One-Cocycles and Knot Invariants by T. Fiedler Vol. 72: Laws of Form: A Fiftieth Anniversary edited by L. H. Kauffman, F. Cummins, R. Dible, L. Conrad, G. Elisbury, A. Crompton & F. Grote Vol. 71: The Geometry of the Universe by C. Rourke More information on this series can also be found at http://www.worldscientific.com/series/skae K E Series on Knots and Everything — Vol. 79 Dennis Roseman University of Iowa, USA High-Dimensional Knotting An Illustrated Guide NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO World Scientific Published by World Scientific Publishing Co.

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