Xjyuk

In knot theory, one can prove that the sum of two knots (say $K =K_1#K_2$) is the unknot if and only if $K_1$ and $K_2$ are both unknots themselves. A proof for this I often hear about is the "Mazur swindle", which has been explained to me as:

$K_1#K_2#K_1#K_2...=K_1#(K_2#K_1...)=0 Longrightarrow K_1=0$.

So both $K_1$ and $K_2$ must be the unknot. I don't understand however, how this works in terms of convergence of this infinite sum. Why is this infinite sum allowed in knots, whereas the seemingly equivalent $1-1+1...$ is not allowed?

Great question! The issue is:

How do we define infinite sums?

Below I've used "$+$" instead of "$#$," to emphasize that the concerns in each case are identical and the only difference is how they're resolved - with inverses not existing in the knot context, and with infinite sums behaving badly in the arithmetic context.

The key point is that in the context of knots, there is a good way to define arbitary infinite sums - where "good" here means that it has nice algebraic properties, and in particular allows the Mazur swindle to go through. It's a bit messy to write down the definition of the infinite connected sum precisely. The simplest approach is to think of knots as continuous injections from $[0,1]$ to the closed unit cube which sends $0$ to $(0,0,0)$ and $1$ to $(1,1,1)$ (intuitively, the actual knot is formed by joining up these two points) and of equivalence of knots amounting to isotopy fixing those basepoints.

Now, we compose two knots intuitively by putting two unit cubes "corner-to-corner," drawing the respective knots in those cubes, and then "scaling down" by a factor of $2$ in each direction. But we can also compose infinitely many knots! Specifically, we start by putting an infinite chain of cubes corner-to-corner, and placing the corresponding knots in each. We then scale down in a more complicated way, as: $$(a,b,c)mapsto (2arctan aover pi,2arctan boverpi, 2arctan coverpi).$$ This fits our infinite concatenation of (shifted) knots into the unit cube; we then add the point $(1,1,1)$ to the whole shebang to get a genuine knot.

The key point is that this is totally well-defined. (Well, it would be if written out a bit more formally, but meh.) Our next step is to rigorously prove things about it; specifically, that it satisfies the appropriate "infinitary associativity law." This isn't hard to do - the isotopy in question is quite easy to write down, if a bit tedious. What we then get out of this is that, for any sequence of knots $(K_i)_iinmathbbN$, the infinite sums $$sum_iinmathbbN(K_2i+K_2i+1)$$ and $$K_0+sum_iinmathbbN(K_2i+1+K_2i+2)$$ are each defined and are equal (fine, they're not literally the same knot, but they represent the same knot class). And from this, we get the Mazur swindle.

Looking at this we can see where the analogous "proof" for arithmetic is incomplete: to finish it, we would need to $(i)$ find a way to assign a real number to every expression of the form $sum_iinmathbbNx_i$, and then $(ii)$ show that that assignment satisfied the appropriate infinitary associativity law. Certainly the usual definition via the limit of an infinite sequence doesn't help us here, since it's not always defined (in particular, $lim_nrightarrowinfty(sum_i=1^n (-1)^i$ does not exist).

Indeed what we learn from the usual paradox is that this can't be done.$^1$ More broadly, we get the general theorem that - roughly speaking - we can never have a context where all infinite sums make sense and behave well, every element has an inverse, and not everything is equal to zero. I don't think this result has a specific name; I've heard it referred to as the Eilenberg-Maclane swindle as well, since it's an immediate fallout of that (if I recall correctly, Eilenberg introduced the same argument in an algebraic - as opposed to geometric - context, at around the same time as Mazur introduced it in knot theory).

$^1$That said, there's lots of interesting math around partial definitions along these lines - that is, notions of "infinite sum" which $(i)$ extend the usual notion to at least some additional series and $(ii)$ have some basic niceness properties. See e.g. here.