# Why are invariants of knots and manifolds important or useful?

It is easy to define invariants that completely classify knots; however, this is computationally infeasible, so is it computationally efficient invariants that are important? Why? Do mathematicians ever sit with two really large knots and want to know if they are the same?

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Why knot?...... – Hayden Feb 18 at 19:31
Mathematicians all the time want to know whether two objects are isomorphic (equivalent, similar etc.) or not. And it even has applications in real life. – Dietrich Burde Feb 18 at 19:32
There are degrees of computationally unfeasible. I would imagine an invariant that is literally uncomputable in general would not be very useful. (Counterexamples?) Even some homotopy groups can be computed. In general, the goal of invariants is to make a hard problem less hard. – Justin Young Feb 19 at 16:26

Personally I often find "They tell things apart!" to be a very uninteresting reason to care about invariants. In all honesty, I am not passionate about whether two complicated enough 3-manifolds (I can write some down if you want) are diffeomorphic except in principle. What I, and I think most mathematicians want, is (to steal the idea from Thurston) understanding of the class of 3-manifolds, or of specific examples.

Except for Lens spaces, irreducible closed oriented 3-manifolds are determined by their fundamental group. This is cool! While one interpretation of this is that we can tell apart 3-manifolds using certain algorithms after calculating their fundamental groups (this is actually possible in this case, as the groups obtained are special), what it tells me instead is "Wow! The topology - and a lot of the geometry, even - of a 3-manifold is captured by its fundamental group." This then leads me to further questions about manifolds and their fundamental groups. (See also Mostow rigidity for a similarly inspiring, and in many ways similar, result.)

So why do I care about invariants? They teach me about the manifolds! The Euler characteristic is multiplicative under finite coverings. Applying this to oriented closed surfaces I learn precisely which surfaces are covers of others. Which $n$-manifolds embed in $\Bbb R^{n+k}$? Stiefel-Whitney classes help me understand this. More specifically, which oriented 3-manifolds embed in $\Bbb R^4$? Well, Stiefel-Whitney classes are no help, but in the special case of homology spheres we invent the Rokhlin or Casson or Froyshov invariants to better understand this question. (In a related direction see Manolescu's work on the triangulation conjecture - he disproved it by constructing an invariant with certain properties.) And suppose I'm really into the geometries of foliations. I learn about the notion of taut foliation on a 3-manifold, and eventually invent the L-space conjecture which relates the existence of a taut foliation to both the fundamental group and the Heegaard Floer homology of the space.

I guess in summary I care about invariants because they prove theorems and teach me new things about manifolds, either in specific inspiring cases or in generality.

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Algebraic topology, and even more so differential topology, are about the mathematical description of "Gestalt". The objects (or spaces, etc.) in question are of a homogeneous, or continuous nature, but we all are convinced that their "Gestalt" is an entity that can be described and classified in a discrete way.

Now knots, resp., imbeddings of $S^1$ in ${\mathbb R}^3$, are "datawise" simple creatures, and they can be presented in a straightforward way on a piece of paper. Therefore they serve as a prime theme (one could call it a "toy model") in this general endeavour that I call "Arithmetisierung der Gestalt" (arithmetization of morphological structure).

Mathematicians do not "sit over knots" with $2869$ crossings; but they want to create tools that distinguish definitively between the knots with $17$ or $29$ crossings that they meet in everyday life, and that might play a rôle sooner or later in string theory. Above all the tools so developed can be of help in classifying geometrical situations of a less intuitional nature than knots in ${\mathbb R}^3$.

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