![]() Haefliger used a higher-dimensional version of the ellipsoidal Borromean rings to construct his exotic smooth embedding of S^3 in S^6, so this is an idea that “has legs.” If the pairwise linking numbers were zero, the discs do not intersect, so shrinking the radius of the sphere produces an animation where the link component radii go to zero, and the link components remain disjoint.Ī corollary of this observation is that the Borromean rings (and the Whitehead link, etc) can not be put into a position where every component is round - this holds true in R^3 as well as S^3, since stereographic projection preserves round circles.Īlthough the Borromean rings can not be realized by round circles in R^3, they can be realized by ellipses. The idea for the proof is that if all the components of a link are round the linking number of components would either be 0 or +-1, depending on whether or not the affine-linear 2-discs they bound in D^4 intersect or not. Given a non-trivial link in the 3-sphere with all pairwise linking numbers equal to zero, it is impossible to put that link into a position where every component is a round circle.ĭefinition: A link in S^3 is “round” if every component is the intersection of an affine-linear 2-dimensional subspace of R^4 with S^3. Specifically, this is an attempt to describe the “spaces of knots” subject in a way that might entice low-dimensional topologists to think about the subject. I want to talk about what I’d call second-order problems in low-dimensional topology, less foundational in nature and more oriented towards other goals, like relating low-dimensional topology to other areas of science. In my mind, the two most representative ones would be the smooth 4-dimensional Poincare hypothesis, and getting a better understanding of the homotopy-type of the group of diffeomorphisms of the n-sphere (especially for n=4, but for n large as well). Most of the foundational problems are solved, and there’s a fairly isolated collection of foundational problems remaining. create the knot with given radius and stepĪs an Amazon Associate we earn from qualifying purchases, so if you’ve got something you need to pick up anyway, going to Amazon through this link will help us keep Hacktastic running.Over the past 10-12 years, geometric topology has entered a new era. take parameterization of torus (u,v)->R^3 remove comments for the one you want to compile tubify module based on tube module from kitwallace #TYPES OF KNOTES THAT HOLD SPHERES CODE#Here is the commented OpenSCAD code that produced the two models in this post: #TYPES OF KNOTES THAT HOLD SPHERES FREE#OpenSCAD is free and makes fairly stable STL files the only penalty is that it is SLOW. In addition, Mathematica does not always export reliable STL files. Making knot models that can export to STL files is easy to do in Mathematica (see Day 110), but Mathematica is not a free tool and not everyone has access to it. Much thanks to kitwallace who had the idea to model curves this way in OpenSCAD. ![]() You can control the thickness of the curve by changing the radius of the spheres. The code below samples points along this curve, puts spheres at those points, and then connects adjacent points with a hull. Technical notes, OpenSCAD flavor: The knots are made using parametric equations that trace out a curve along the surface of a torus. ![]() If we substitute u=pt and v=qt then as t runs from 0 to 2휋 we obtain a curve that traces the T(p,q) torus knot around the surface of the torus (and if p and q are reversed then we get the T(q,p) torus knot). ![]() The parametric equations for a torus with handle radius a and large radius c are (see Wolfram): x = (c + a cos v) cos u ![]()
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