A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$. Ambient isotopic braids are considered equivalent. It makes no difference whether or not the ambient isotopies are required to be level-preserving, i.e. that two braids are `level-preserving ambient isotopic' if and only if they are ambient isotopic. This is a non-trivial result.

A string link is a smooth (perhaps not level-preserving) embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$. Ambient isotopic string links are considered equivalent. Note that components of string links may be knotted, but components of braids cannot be knotted.

A tangle (for purposes of this question) is a string link in which some components might start and end at the top $\mathbb{R}^2\times \{1\}$ or at the bottom $\mathbb{R}^2\times \{0\}$ which may also have closed components (knots) $S^1\hookrightarrow \mathbb{R}^2\times [0,1]$, also all up to ambient isotopy. In particular, a knot or a link is a tangle.

Braids inject into string links which inject into tangles. Tangles are what you get when you slice knots and links into pieces, hence people who study knots and links care about them. Usually invariants are first constructed for braids (where they are relatively easy), then for string links, then for tangles. For example, the Kontsevich Invariant was first constructed by Kohno for braids (where it is computable) and then by Kontsevich for tangles (where it is complicated).

Sometimes in mathematics, embedding one class of objects inside a `completion' helps us to study the smaller class. For example, many facts about integers are proved using real and complex numbers as discussed in this MO question. My question is whether the study of tangles (or string links) has given us any insight into the theory of braids- whether relaxing the level-preserving requirement (that $f(k,t)\in \mathbb{R}^2\times \{t\}$ for all $t\in [0,1]$) leads to results about braids.

More Specific Question: What do tangles teach us about braids? If we were interested only in braids, would we have reason to introduce string links and tangles?
More General Question: What do pseudoisotopies teach us about isotopies? If we were interested only in isotopies, would we have a reason to introduce pseudoisotopies?

I asked a narrower MO question a few months ago and the crickets chirped; I'm still very much interested in understanding what is going on here, because it seems to me to shed light on the core of the interface between braid theory and classical low-dimensional topology.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.