M Theory Lesson 82
Remember the Lorenz template? Now Terence Tao has a fantastic post bringing our attention to the work of Etienne Ghys.
Ghys does amazing things with knot dynamics for the space of (unimodular) lattices in $\mathbb{R}^{2}$, which is the complement of the trefoil knot in the 3-sphere. Elements of the modular group define a periodic orbit in this space, and Ghys shows that this class of knots is the same as the Lorenz knots! And the linking number between such a knot and the trefoil is the Rademacher function $R$ from SL(2,$\mathbb{Z}$) to $\mathbb{Z}$ defined by
$2 \pi i R(A) = 24 (A(\tau) - \tau) log \eta - 6 log (- (c \tau + d)^{2})$
where $A = (a,b;c,d)$ and $\eta$ is the Dedekind function! Carl will like those 12th roots of unity again. The proof of similarity to the Lorenz system uses hexagonal and rhombic tilings which are deformed towards the Lorenz template.
Ghys does amazing things with knot dynamics for the space of (unimodular) lattices in $\mathbb{R}^{2}$, which is the complement of the trefoil knot in the 3-sphere. Elements of the modular group define a periodic orbit in this space, and Ghys shows that this class of knots is the same as the Lorenz knots! And the linking number between such a knot and the trefoil is the Rademacher function $R$ from SL(2,$\mathbb{Z}$) to $\mathbb{Z}$ defined by
$2 \pi i R(A) = 24 (A(\tau) - \tau) log \eta - 6 log (- (c \tau + d)^{2})$
where $A = (a,b;c,d)$ and $\eta$ is the Dedekind function! Carl will like those 12th roots of unity again. The proof of similarity to the Lorenz system uses hexagonal and rhombic tilings which are deformed towards the Lorenz template.
posted by Kea | 1:06 PM
2 Comments:
Really amazing. I do not have a slightest idea about how they prove these results. The post of Terence Tao is fantastic service to non-professionals.
Tao mentions helicity as an invariant of fluid flow. (Abelian) Chern-Simons action for lightlike 3-surfaces has interpretation as helicity when Kahler vector potential is identified fluid velocity. It seems that it is flow flows of A, rather than B which define naturally braids.
Helicity is not gauge invariant and this is as it must be in TGD framework since CP_2 symplectic transformations induce U(1) gauge transformation which deforms space-time surface an modifies induced metric as well as classical electroweak fields defined by induced spinor connection. Gauge degeneracy is transformed to spin glass degeneracy.
It is amazing that very probably single light-like 3-surface could realize all possible braids!
Hi Kea, Matti,
The Tao thread of Ghys' work is really fascinating.
I made one of my speculative comments attempting to relate this to the work of others that may be similar.
I do tink that there may be a possibility that the helicity may contribute to solenoid like activity.
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