Loops without any crossing left deserve the same role the circle has got in the standard plane, so we might compute polynomials relative to our graphs. That is to say the Reidemeister movements are (discretely) done on our manifolds and not in the projection. And (we hardly dare to guess, but) of course those movings ropes around (without cutting) form some group to be investigated.
Using again the ExportData utility (-kd and -kp) you get the path data and (by the help of our preferred computer algebra system afterwards) the generalized Jones polynomial using the writhe and some skein tree. The definition is as follows: Let L denote the general link, and <L> the polynomial. The rules below can be applied (making the polynomial X (but not L) invariant under the three Reidemeister movements) (please excuse the ASCII art):
Rule 1: <o> = 1
Rule 2: <'\'> = A <=> + A-1 <||>
Rule 3: <L o> = (-A-2 - A2) 1
A crossing like w is the sum of all signatures.
The Jones (a special case of the more general Kauffman) polynomial (btw. failing to be a complete invariant able to distinguish between all different knots) is retrieved by adding the factor
X = (-A3)-w <L>
,and finally a substitution is done:
t := A-4
In order to do those polynomials in our graphs (instead of on the standard plane or cube, that is to say on s.th. with genus 0) we modified the rules a little bit. We apparently have to specify some more atomic results in addition to the rule one, for a loop on a torus without crossings may represent a trefoil and not the trivial unknot. In rule 3 (replacing the 1 there) a multiplication with the polynomial of the loop should take place. And the overall writhe should on one hand consist of the things visible on the very graph itself as well as the parts belonging to the loops. We lose crossings in between different loops (in skein tree and writhe) (and this may actually be the gain of the task and the reason to do the procedure on a generalized graph).
On genus 0 graphs we get the standard results of course. See for example the 623 Borromean rings with writhe 0 (yes, it looks a little bit strange, we admit that) with the polynomial V(t) (lowest exponent and coefficients): {-3}[-1,3,-2,4,-2,3,-1]
Let us take an easy example now (a self-brewed link with one crossing, the skein tree stops at a trefoil and a (1,1) unknot) to speak about the more general results. Here is a screenshot:
We paid attention to the fact that paths without any remaining crossings on a torus can in fact be different from the trivial unknot, so we left those loop polynomials and the writhes as variables to be set manually (inserting -A-5 - A3 + A7 for the trefoil L0 and -3 as its writhe w0 contribution). It seems that a (1,1) torus knot (which is actually standing alone the unknot as well) should get s.th. like A (and not simply 1) and writhe 0. Then we get as polynomial V(t):
{5}[1,-1,1,-1]
and
{-9/2}[-1,0,0,1]
in the other direction respectively replacing all A with its inverse.
Otherwise sometimes some unclear things are going on in the procedure, leading to broken exponents even for initial knots (for links this may often happen) and loosing of coefficients in the final crude substitution.
Probably all this has to do with the chains occuring in Chapter 9, Algebraic, Difference and Geometric Topology, the cycles becoming our entites and the boundaries are shrinkables (and building the quotient we arrive at residues?!). Or it's a local ring?!
Anyway the goal should be to retrieve the same results as if we would do on genus 0, or perhaps the polynomials created on general graphs have their own meaning? Well, this will perhaps be clarified in the future (in any case they should be invariants).
Exercise (worth 5 points): Compute the Jones polynomial of the colorful flower power knot in Figure 10.5, “Hexa Tor Knot”.
Figure 10.5. Hexa Tor Knot
"ab" knot cover with 578 crossings (?!), writhe 67 (?!), 6 levels on "Hexa Tor" graph of genus 2
Instead of using constant word ("ab" or "aab") one may construct paths using a constant acceleration or other things. The methods could also be applied to links in case of multiple pieces walking around. Besides there are other polynomial invariants (HOMFLYPT etc.) to be dealt with.
Vertex operator algebras, Hecke algebras as well as quantum groups (algebra with addition, product and coproduct) (quantized enveloping algebras of semisimple Lie algebras) might give further insights?!