M Theory Lesson 239
Truncating the braid group $B_{3}$ on three strands at a cubed root of unity is a little bizarre, because it leads to equivalences which do not preserve the number of loops in a link. Most notably, since each generator satisfies a rule $\sigma^{3} = 1$, there is an identity taking the trefoil knot on two strands to two straight strands. However, we knew that the Jones polynomial for the trefoil knot,
$J(t) = t + t^{3} - t^{4}$
evaluated at a cubed root of unity, is equal to 1. This is also the value for the unknot. There are 24 elements in the truncated group. More general torus knots have polynomials that may be normalised with respect to the trefoil polynomial, which is another way of setting $J$ to 1.
Aside: For the even simpler case of $B_{2}$, forcing the double crossing to be the identity sends the Hopf link polynomial to $2i$ at $t = -1$, or to zero at $t = i$.
$J(t) = t + t^{3} - t^{4}$
evaluated at a cubed root of unity, is equal to 1. This is also the value for the unknot. There are 24 elements in the truncated group. More general torus knots have polynomials that may be normalised with respect to the trefoil polynomial, which is another way of setting $J$ to 1.
Aside: For the even simpler case of $B_{2}$, forcing the double crossing to be the identity sends the Hopf link polynomial to $2i$ at $t = -1$, or to zero at $t = i$.
posted by Kea | 8:32 PM
4 Comments:
Dear Kea,
I read an article in New Scientist about knot invariants. This inspired what looks like an idea- maybe a trivial one.
The homotopy group of complement is knot invariant. The question is whether one could define braided version of this group and perhaps define new Jones polynomial like knot invariants as quantum traces of unitary quantum group representation matrices for the fundamental group too.
At least at first glimpse this seems be possible. One can assign to a braid a knot and more generally, links by joining the upper and lower ends of strands suitably. The upper end of n:th to lower end of n+1:th for instance cyclically gives a knot. The other extreme is N-link for N-braid. I do not remember how much non-uniqueness this representation involves and whether it exists always for a given knot as would seem.
Let g be an element of fundamental group represented as a link associated with knot. One can cut this link in the region outside the braid and join the ends to the upper and lower ends of braid defining the knot so that a link results when the ends are connected outside the link in such a manner that no new linking or knotting results.
This assigns to the knot and corresponding N-braid an N+1-braid and one can define braid/link invariant as quantum trace. In this manner all elements of the fundamental group of complement (the number is infinite) are represented and define new quantum invariants for the original knot.
Physical picture suggests that one could add also several loops representing elements of fundamental group in this manner so that infinite number of additional Jones invariants telling how these many particle states feel the presence of the braid defining the knot (or link).
The extreme situation would be Jones polynomials for all braids containing the braid defining the original knot.
The natural restriction would however be that the braid defined the added elements of fundamental group is trivial so that minimum amount of additional structure is brought in. I do not know whether this trivializes the situation.
Large, perhaps even infinite number of braids would define braid/link/knot invariants for given braid. This would seem to conform with category theoretical thinking and also with physicist's manner to get information about a physical system by perturbing it: now by adding the links representing elements of fundamental group.
Probably this is just a trivial point in which case I can only apologize for my deep ignorance.
I wrote a little posting about quantum invariants defined by elements of fundamental group of the complement of knot.
Thanks, Matti. These thoughts are appreciated. I am very slow to figure things out, but I do suspect that there must be some nice way like this to create knot invariants.
Dear Kea,
the idea relies on the possibility to represent knots and more generally n-links as braids.
The idea as such is childishly simple and it is difficult to believe that it might have remained un-noticed by professionals in the field.
Transform a loop representing an element of homotopy group of complement of n-link to a new braid strand.
The only reason why this idea might have remained un-discovered is that mathematician usually tries to construct braid invariant using only the data about braid itself rather than data for braids containing it. For a physicist the idea about describing braid as sub-braid affecting the rest of braid is natural. After all, all that we know about electron is through interactions of electron with the rest of the Universe!
In fact, Vassilev takes this dynamical view to some degree. He constructs his invariants by deforming the singular knots by making them non-singular in both possible ways at each singularity.
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