tally

Part II, One Graph, Four Covers · defined in Chapter VIII, Space as a Tally

The net signed number of crossings of each letter in a history of re-anchorings, with the order forgotten.

Plate W.18The kernel graph with the tree letters at report 0 and the chords 12, 13, 23 counted by

z_1

z_2

z_3

; the triangle

0\to1\to2\to0

, in gold, closes in

K_4

and crosses one chord.

Write $e_{ab}$ for the letter $ab$ crossed from $a$ to $b$ , with $e_{ba}=-e_{ab}$ . The tally of a walk is the sum of its crossings, an integral one-chain whose boundary is the endpoint minus the start. A closed walk has a tally in $H_1(K_4;\Z)\cong\Z^3$ , with the three triangles through report 0 as a basis, so the tally of a walk from 0 is a pair $(v;z)$ : the report it ends at and the net crossings of the three chords 12, 13, 23. The triangle $0\to1\to2\to0$ ends at $(0;1,0,0)$ , back at report 0 but not at the start, and the two orders of two triangles and a square all end at $(0;1,0,1)$ .

The tally forgets order exactly: two histories from one report have the same tally when they differ by a commutator of $F_3=\pi_1(K_4)$ . It separates all histories of length at most four, and the first pairs it identifies have length five and close a decagon of the crystal. What it forgets, the kernel $[F_3,F_3]$ , carries the non-abelian part of every holonomy.

Proposition(Circulation coordinates) proved

$H_1(K_4;\Z)$ is free of rank $6-4+1=3$ , with basis the three triangles through report 0, $c_1=e_{01}+e_{12}-e_{02}$ , $c_2=e_{01}+e_{13}-e_{03}$ , $c_3=e_{02}+e_{23}-e_{03}$ . Every walk from 0 ending at $v$ has a tally $t_v+c_1z_1+c_2z_2+c_3z_3$ with $z\in\Z^3$ , where $t_0=0$ and $t_v=e_{0v}$ otherwise. The integers $z$ are the net signed numbers of crossings of the three letters 12, 13 and 23.

Proof

Take the three letters at 0 as a spanning tree. Each of the remaining letters, the chords, occurs in exactly one $c_i$ with coefficient one. Subtracting those chord coefficients from an integral circulation leaves a circulation supported on the tree, which vanishes: a leaf’s boundary equation forces its edge coefficient to zero, and induction removes the tree. For an open walk, subtract $t_v$ first.

As mathematics

For a walk $v_0,v_1,\dots,v_n$ on $K_4$ , the one-chain $c=\sum_ie_{v_iv_{i+1}}\in\Z^6$ , the signed Parikh vector of the history’s word. For a closed walk it is the class of the loop in the abelianization $H_1(K_4;\Z)\cong\Z^3$ of $\pi_1(K_4)\cong F_3$ , the Hurewicz map; with the start, the tally of an open walk is its endpoint in the maximal abelian cover of $K_4$ , the crystal.

Its name in another field	Bridge
the signed Parikh vector of the history’s word	built
a class in $H_1(K_4;\Z)$ , the abelianization of $F_3$	built
with the start, a vertex of the $K_4$ crystal	built
a position	a reading

Built from: letteranchorkernel graph
Builds: crystalkernel
In the volume: IIIThe KernelVFour Reports, Six LettersVIIThe Branchial TreeVIIISpace as a TallyIXWhat Space ForgetsXIRulial RelativityXXThe Commuting SquaresXXIIILight, Vacuum and Handedness