Why do we study representation of knot groups?

Question

In my independent study class in Knot Theory, my professor said that solving the relation in knot groups is an undecidable problem, and he said we study representations of knot groups because then solving the relation is easier. I don't know much about representation theory yet, but I would like some insight on why do we study representation of knot groups and the current research carried in that direction.

Answer 1

For any group $G$, consider the following two problems:

1. For some explicit $x,y\in G$, prove that $x=y$.
2. For some explicit $x,y\in G$, prove that $x\neq y$.

If your group is given as a subset of some set with decidable equality - e.g., any group of matrices - then both problems are easy: In the case of matrices, for example, just check if all matrix entries are equal or not.

However, if your group is given abstractly through generators and relations, then while (2) has a simple notion of proof (namely, a set of transformations from one element into the other by means of the given relations), how would you prove (1)? There are infinitely many ways in which you can apply the relations, and it requires some additional idea/argument to come up with an argument that, however you apply the relations, you'll never be able to transform the presentations of the given group elements into each other.

You can view a representations as a generic such argument: They allow to solve (2) by reducing it to the case of 'concrete' groups where we can solve it easily by inspection: Indeed, a representation is just a homomorphism $\varphi: G\to \text{GL}_n(K)$ into some matrix group over some vector space, so in order to show $x\neq y$, you it suffices to find a $\varphi$ so that $\varphi(x)\neq\varphi(y)$.

All this is very general, but actually, in knot theory you see something very similar: It's easy to convince yourself that two knots are isotopic -- just give me a deformation of one into another -- but how do you prove that two knots aren't? Again, you have infinitely many ways in which you could deform the knots. Analogously to using representations to map abstract groups to something concrete that can be easily compared, knot invariants also assign something 'concrete' (such as a number or a polynomial) to knots in an isotopy-invariant way, so you can prove non-isotopy through in-equality of the invariants.

Answer 2

Representations onto abstract groups allow the explicit construction of covering spaces associated with the various permutational representations of such groups. See Heegaard's 1898 Thesis. This yields lots of simple homology and linking invariants of knots -- enough to distinguish all of the billions of examples encountered thus far. See Thistlethwaite's tables. 3-colorable knots (those whose group maps homomorphically onto the symmetric group on three letters) are particularly easy to deal with. See our Researchgate project.