What Makes a Knot Knotty?
Can it be just the not-ation?
Colin Rourke is a topologist at the University of Warwick in England. He is one of numerous mathematicians whose work is surveyed in an entertaining article by Sam Nelson of the Claremont Colleges in California, “The Combinatorial Revolution in Knot Theory” which appeared in last month’s AMS Notices. Rourke’s home page has interesting materials and notes besides a selection of his papers. These include a one-page formulation of Gödel’s first incompleteness theorem and a model for the universe that “overturns nearly every tenet of current cosmology.”
Today Dick and I wish to draw attention to the mathematical structures in Nelson’s article, whose properties have been developed by Rourke and others, and to how syntax may take precedence over the ‘real physical meaning’ of a mathematical concept.
Nelson summarizes the “revolution” in his opening paragraph as follows:
Instead of thinking of knots…geometrically as simple closed curves in , a new approach defines knots combinatorially as equivalence classes of knot diagrams under…certain diagrammatic moves.
Much as the concept of number needed to move from ‘real’ to ‘imaginary’ in order to yield the fundamental theorem of algebra, so are the more general not-ation based not-ions of knots needed to fill out invariants of certain mathematical structures.
Rourke also made an important advance on the Poincaré Conjecture in 1994, about a decade before it was completed to a theorem by Grigory Perelman. He proved that there was an algorithmic procedure that would generate a counterexample to the conjecture if it were false. In logical terms, he proved the negation of the conjecture equivalent to a , i.e. existential, statement of arithmetic. Of course Perelman’s proof obviated this theorem. However, the idea matters to us because is only known to be a statement. Proving it equivalent to a or statement, so that at least one of the ‘equal’ or ‘not equal’ cases would be provable, would be the greatest development in decades.
Knots Into Codes
According to a multiple-sourced legend, Alexander the Great was once challenged to untie a complicated knot that had tied the ox-cart of a peasant-king named Gordias for many years. Alexander either used his sword or pulled out the rod of the cart by which it was tied, thus creating or exposing ends of the string by which to undo it. Pulling out an end of a string to unravel a knot may not yield any useful mathematics, however.
Alexander could have first fluffed the knot so that no more than two segments of string would cross each other at any point. Then the knot’s shadow yields a planar graph of degree 4 in which each vertex has two “over” edges and two “under” edges, and this is the knot’s diagram. If the knot had eleven crossings, the diagram might look like this:
If we trace the diagram beginning at the arrow, assign a number for each crossing in the order encountered, and use a minus sign when traversing an “under” edge, we can abstract the knot into the sequence of numbers
Of course this code is named after Gauss. There are several variations on it. We adopt the “signed Gauss code” in one of Nelson’s attractive knot-theory pages, but using to mean “under” and suggest complex conjugation—in line with a footnote in his survey.
To define it formally, choose as the principal eighth root of unity. The crossing elements are now subscripted copies of a power of . If you are following the “over” edge with the “under” edge directed right-to-left, take ; if left-to-right, take . If you are following the “under” edge with the “over” edge going right-to-left use ; if left-to-right then by . If the second crossing has this last case we can write as short for . The code always has a where the original Gauss code had a sign. The above knot gives:
Flipping the knot over conjugates every element, while any reflection (right to left or top to bottom) negates every one. Reversing every edge reverses the sequence but does not change any or . As with the “extended Gauss code” defined in our source for this diagram, this code preserves enough information to reconstruct the knot from the diagram.
The code yields the trefoil knot. Note that in any code, each element appears once with a and once without. Thus the codes are some kind of generalized commutator.
Diagrammer Move Grammar
Suppose the modified Gauss code has the sub-sequence
for some crossing element . This means we’ve drawn a loop over or under in he direction of travel, but the loop isn’t wound around anything. So it can just be pulled taut. Since the crossing elements are complex units, we already have . Note that since the codes are really cycles, and are the same cases.
Now suppose instead that the modified Gauss code has
where and have the same sign, and the subscripts on and differ by . In words this means we have two consecutive over-crossings of the same other segment, or two under-crossings of it. In either case the two segments are merely sitting one atop the other and are not linked. Again these sub-sequences can be erased.
To describe the third move, suppose we originally had 3 segments cross at one point: a lower strand going northwest–southeast, a middle strand going southwest–northeast, and a top strand going east–west. When Alexander fluffs the knot, the top segment can either bend up or bend down. The two resulting equivalent configurations are:
Suppose the topmost segment is first traversed west-to-east, next the bottom one NW-to-SE, and last the middle one SW-to-NE. The corresponding rule for the modified Gauss code is:
There are similar rules for the other arrow directions and orders of visiting the segments. The diagram comes from this article in John Armstrong’s blog “The Unapologetic Mathematician” which we keep in our sidebar, and shows colorings associated to diagrams.
How Knotty is the Not-Knot?
Nelson’s article shows the moving strand in the middle rather than on top. This doesn’t matter—if we were to move the SW-NE strand in the left diagram northwest, we would still get the right diagram. This neat theorem was proved in 1923 by Kurt Reidemeister:
Theorem 1 A knot diagram represents the unknot if and only if all of its crossings can be eliminated by some sequence of the above moves—equivalently, iff its (modified) Gauss code is reducible to 1.
As shown by Dick’s posts on work with Zeke Zalcstein, we are interested in complexity properties of these kinds of rewriting systems. In particular, the problem of whether a diagram represents the unknot is in , and is in if the Generalized Riemann Hypothesis is true, but has not been proved to be in . GRH is needed only to get good bounds on primes such that certain integer polynomials associated to the knots have roots in .
To Knot or Not to Knot, or N-Knot?
The point of departure in Nelson’s survey is that the rewriting theorems apply even to codes that are not from shadows of “real” knots in 3-space. The simplest such code is the commutator
Any shadow that includes this pattern must have at least one more crossing. However, a diagram giving this code can be drawn on the torus, looping from under and around to join the line into .
His survey goes into a concept of virtual crossings and virtual knots introduced in this 1999 paper by Louis H. Kauffman. There is no Gauss code for the virtual crossing itself, though we have marked it in this diagram:
The point is that every (modified) Gauss code—subject only to the rule of every element and its conjugate appearing once each—has a planar diagram with virtual crossings. This allows spatial intuition and properties about crossings to be applied to all codes.
In passing we ask whether there is any utility in considering in the diagram to be a “nondeterministic crossing” instead—with either strand possibly being the “over” one. This could be coded as
Note that if the SW-to-NE strand in crossing is “over,” then we get the trefoil knot. If “under,” then it becomes which the second Reidemeister move applied to transforms to . (One could also use the move for since it is a cycle.)
Another way to think of an “N-Knot” is that for any element labeled nondeterministic, it is legal to swap and . Is it harder to decide whether an N-Knot can be reduced to than for a regular (or virtual) knot? This is different from computing the unknotting number insofar as only some crossings are nondeterministic. It strikes us that this problem ought to be -complete, which would make it unlikely that an efficient invariant can be found to prove that a given non-trivial N-knot does not reduce to .
Not Knot-Code Codes
It would be nice to avoid the non-local coupling between the two times a crossing is visited. Therefore let us try to abstract the crossing itself. Or, let us abstract the notion of a segment as it goes through one or more crossings. Define to mean that goes under at a crossing. Optionally we can specify to mean that the crossing is positive in our signed Gauss code, and use when it is negative. The three Reidemeister moves then yield these axioms:
That and the last equation with in place of everywhere follow from this. When the distinction between and is ignorable, these define a structure called a kei, else they define a quandle. These rewriting systems tell some already-familar stories.
Keis and Quandles and Racks, Oh My
Here are two examples that distinguish a kei from a quandle. Consider any group, such as one of nonsingular matrices. Define:
The first operation gives a kei because and
The latter two operations give a quandle and not a kei because both are needed to cancel the ‘s, namely
We can get the same examples with unitary matrices and for inverses, and this is just one hint of why knots and keis and quandles are relevant to quantum mechanics.
If a quandle is “excused” from obeying the first axiom, then it becomes a rack. Our last example, from Nelson’s survey and quandle page, shows all three. Consider any algebraic structure that admits left-multiplication by objects and , such as a module. For define:
The middle two operations define the Alexander quandle, which needs no modulus, while the first defines only a rack because makes no assumption on .
David Joyce showed in his dissertation in 1980 that two oriented knots are related by a homeomorphism of the 3-sphere if and only if the quandles they generate are isomorphic. Racks go with a further loosening of the idea of knots, and working with Roger Fenn, Colin Rourke proved a similar theorem. Nelson’s survey closes with links to further applications. We are interested in what computational complexity may say about them.
Can we extend the applications of these knot-derived notions?
Is the set of (codes for) unknots in ? Does membership in need GRH?
Are there coding schemes that are both smarter in helping prove theorems and simpler in complexity?
[changed way signs were described in second move; inserted “negation of the” before Poincaré conjecture.]