![]() These subjects study materials and their behaviours under different stresses and conditions. Specifically, continuum mechanics: subjects like elasticity and plasticity. One of the natural reasons to study spaces of knots comes not from foundational 3-manifold theory questions, but from mechanical engineering (considered broadly!). So in this topology two such embeddings are close only when there is a “small” isotopy from one to the other. That’s the topology where one takes as a distance between two smooth embeddings f,g : S^1 –> S^3 the maximum of |f(z)-g(z)| + |f'(z)-g'(z)| where z is in the circle S^1, it is sometimes called the Whitney Topology. In spaces of knots, the objects of study tend to be things like the space of all C^1-smooth embeddings S^1 –> S^3 with the C^1-metric topology. Cerf theory, sweep outs, singularity theory, open book decompositions and Teichmuller theory all have aspects of the spaces-of-things philosophy, where one studies families. ![]() There are times when that’s less of the case. These are relatively discrete-ish problems. Various cobordism relationships between manifolds, surgery relationships, and so on. Much time has been spent in geometric topology on relatively foundational problems, like classification problems. ![]() You might ask “what does all this have to do with spaces of knots?” It’s about time we got to that. Here is one elliptical embedding of the Borromean rings in R^3: 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. Over the past 10-12 years, geometric topology has entered a new era.
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