Analytic Representation of Generalized Möbius-Listing’s Bodies and Classification of Links Appearing After Their Cut

Pinelas, Sandra; Tavkhelidze, Ilia

doi:10.1007/978-3-319-75647-9_38

Cited by 5 publications

(11 citation statements)

References 7 publications

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“…Solutions for this problem have been obtained for GML surfaces and bodies with m = 2, 3, 4, 5, 6 [1][2][3][4][5][6] and revealed connections with the field of knots and links, depending on the number of twists of the original cylinder or prism with m-symmetrical cross section, given by the superscript n in GML n m . The method of study was the full cutting of GML n m surfaces and bodies with a moving knife in 3D.…”

Section: Propositiomentioning

confidence: 99%

“…The method of study was the full cutting of GML n m surfaces and bodies with a moving knife in 3D. More general solutions were obtained when the cross section of the GML n m surface or body is a Gielis curve, describing the boundary of the cross section as well as the disk [4][5][6][7]. Gielis curves can transform the cross section of the GML n m from a circle to a regular polygon and vice versa, and to many other concave or convex curves.…”

Section: Propositiomentioning

confidence: 99%

“…Thus far, the classification of cutting of GML n m surfaces and bodies was achieved for classic Möbius bands [3], and for GML n m with m = 2, 3, 4, 5 and 6 [4][5][6]. A full classification was achieved for cutting of classic Möbius bands with any k number of knives [3].…”

Section: From the Original Motivation To The General Problemmentioning

confidence: 99%

“…Indeed for 3D GML surfaces and bodies, the shapes can be very complex, and a full classification has been achieved for m = 2, 3, 4, 5, 6. The case 2 refers to ribbons, but can also be 2-angular figures [1][2][3][4][5][6].…”

Section: Gml Bodies and Relation To Knots And Links After Cuttingmentioning

confidence: 99%

“…The identification links planar geometry to 3D GML n m bodies. If one follows the d 1 -knife along the basic line of the GML 6 6 body, the different positions of the knives indicated by arrows in Figure 44 for d 1 function as a clock, relative to the torus circumscribing the GML 6 6 body. For other knives the clock arithmetic is the same, albeit with more hands.…”

Section: Demonstratio the Geometrical Solutionmentioning

confidence: 99%

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The general case of cutting of Generalized Möbius-Listing surfaces and bodies

Gielis

Tavkhelidze

2020

4open

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The original motivation to study Generalized Möbius-Listing GML surfaces and bodies was the observation that the solution of boundary value problems greatly depends on the domains. Since around 2010 GML’s were merged with (continuous) Gielis Transformations, which provide a unifying description of geometrical shapes, as a generalization of the Pythagorean Theorem. The resulting geometrical objects can be used for modeling a wide range of natural shapes and phenomena. The cutting of GML bodies and surfaces, with the Möbius strip as one special case, is related to the field of knots and links, and classifications were obtained for GML with cross sectional symmetry of 2, 3, 4, 5 and 6. The general case of cutting GML bodies and surfaces, in particular the number of ways of cutting, could be solved by reducing the 3D problem to planar geometry. This also unveiled a range of connections with topology, combinatorics, elasticity theory and theoretical physics.

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