Topologically, what is a 'string' from string theory?

To begin: I am not a crank. I am not sure how well-founded my titular question is, but it was interesting enough that I decided to bring it to MSE.

For context: I am an undergraduate mathematics student. I am taking a course in algebraic topology, and during class it was asked how algebraic topology applies to string theory. Our professor explained that he was not sure, but he knew there was some connection between knot theory and string theory. He said it may interesting to understand the topological role of a 'string,' and I assume this is contingent on the connection between knot theory and string theory being strong enough that such an analogy can be realized.

So I am here to ask:

1. Is the connection between string theory and (algebraic) topology strong enough that it makes sense to ask how a 'string' can be viewed topologically?
2. If so, what is a string from this topological point-of-view?

The strings in string theory are just one-dimensional manifolds embedded in a (usually) 10-dimensional space. The strings represent particles; the basic idea is to replace the pointlike concept of a particle in classical and (though it is more complicated here) quantum mechanics with a particle that is stringlike. This gives particles much more interesting behavior, such as the ability to have discrete vibrational modes. As for the strings themselves, there isn’t really a deeper connection to topology that I am aware of.

However, there are important connections between string theory and algebraic topology. This comes up in several places. I am not a string theorist, so I cannot enumerate all of these, but the main one has to do with the fact that string theories live in more than three spatial dimensions (most commonly ten). Since we only observe three dimensions, you have to explain why we don’t see the others. The oldest mechanism for this is Kaluza-Klein compactification, where the extra dimensions are like a compact manifold that is very small in size. The manifold has to be a Calabi-Yau manifold, and topology is highly relevant to this part of the theory.