This post has comes with a PDF version, which contains more details.
In my opinion, Habiro and Goussarov’s theory of claspers is a milestone in low dimensional topology . Using this theory, they were able to solve important outstanding problems in quantum topology, and to build a bridge to connect the “quantum” with the “classical”.
In this series of posts, I would like to introduce claspers to readers of this blog. I think that claspers are an handy tool for low-dimensional topologists, as a natural extension (or refinement) of Dehn surgery which (in a sense to be explained in future posts) is compatible with the lower central series of the Torelli group of a Heegaard surface. Claspers are already fairly mainstream in some circles (used or referenced in around 300 papers according to MathSciNet), but I think that they should be more popular with a wider audience (if gropes are familiar to you- in dimension 3 claspers and gropes are essentially equivalent).
Claspers are one of those good ideas which are useful even when emptied of their content, when only the shell remains. In this first post, I would like to ignore the content— all the beautiful general constructions and shimmering theoretical triumphs of the theory— and focus instead on just (a small part of) the form—claspers as notation. Just graphical notation and nothing else! Look how efficiently it presents mathematics in your head!
My example today is a proof of the following theorem (explained below) :
Theorem: Two links are link homologous if and only if they are related by -moves.
Two oriented and ordered links and are said to be link homologous if and for all and with . The terminology is explained by the fact that the linking number is defined as the image of the th longitude in , so (loosely speaking) two links are link homologous if and only if their homologies agree. The -move is a certain local move we will soon define. Thus the theorem characterizes -move equivalence classes (geometric topological information) in terms of a readily computable algebraic invariant.
The proof which I present below, in bold font, was told to me by Andrew Kricker last year in under 30 seconds, who had heard it from Habiro, who was reformulating Murakami and Nakanishi’s original argument, which was no doubt known to Matveev in his original paper where he introduce -moves  (It may appear explicitly in a 1986 Matveev preprint “Nonstandard surgeries on 3-manifolds”— I don’t know because our library does not seem to have a copy. Please tell me if you know). This then is math folklore.
Let’s introduce our graphic notation tool, which is a special kind of basic clasper which I’ll call “chord claspers” (just because I’m biased against non-descriptive adjectives like “basic”). A chord clasper in the complement of a link is the union of two annuli and called leaves whose deformation retracts (which are circles) bound disks each of which intersect L at precisely one point, with a band B which connects them, which is called an edge. In this post I’m going to assume that the leaves are not twisted. Claspers may be drawn using their cores by the following rewriting rules:
Chord claspers are nothing more than graphical shorthand for linkage:
They look just like kayak paddles!
It’s time now to introduce the -move (in a few versions):
As proven in the PDF version of this post, these moves are equivalent, and we collectively call them the -move.
Here is the proof in bold font, with my explanations in normal font:
Undo the link, replacing it with an unlink and some messy web of chord claspers hanging between the components.
Any link may be undone by crossing changes, which in turn may be realized by introducing chord claspers.
Realize “permuting leaves” and the clasp-pass move using \moves; and untwist edges
Permuting leaves is , which according to our definition is the -move. The clasp-pass move is the following move:
We untwist edges as follows, using :
Now the whole messy web comes undone, and you are left with the “standard dude”.
Choose a “standard dude” , which is a pure braid with the property that for each and with .
The link may be presented as the closure of a tangle . Stack and above :
The linking number of any two arcs in vanishes. When working modulo -moves, edges do not see other edges, and leaves do not see other leaves. Edges don’t see the link either because of . They don’t see anything at all. So we can rearrange claspers at will. If both leaves of some chord clasper clasp the same component , use and clasp-pass repeatedly to get the local picture below, and remove the clasper:
So all chord claspers both of whose leaves clasp the same component in can be killed (in particular, the -move is an unknotting move). What about the ones whose leaves clasp different components? Well, the number of half-twists in the edge of a chord clasper can be 2-reduced as we saw above, and two chord claspers between and whose edges have numbers of half-twists with different parity cancel.
This cancels all chord claspers between and in because of the condition that .
And voila, comes undone, and you are left with the standard dude . Thus any two link homologous links are -move equivalent to and therefore to one another.
We’re done. Any questions?
Edit: Does anyone have any idea why line 8 of page 3 of Naik and Stanford’s paper states that the proof only “appears to generalize” from links to string links?