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~ ~ ~ ~ ~ ~ ~ ~
~ ~ + -- -~ ~
+
~ ~ +
+
-
-
-11 0~ ~ +
+
+
2 11 = [+1 / 2, 1/ 2, 1/ 2]P 23 [+1 / 2, +1 / 2, 1/ 2]P 12= [+1 / 2, 1/ 2, 1/ 2][+1/ 2, 1/ 2, +1 / 2]P 23 P 12
= [+1 , 1, 0]P 23 P 12= [+1 , 1, 0]P (132)
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( 2 11 )
3 == [+1 , 1, 0]P (132) [+1 , 1, 0]P (132) [+1 , 1, 0]P (132)= [+1 , 1, 0]P (132) [+1 , 1, 0][ 1, 0, +1] P (132) P (132)
= [+1 , 1, 0]P (132) [0, 1, +1] P (132) P (132)= [+1 , 1, 0][ 1, +1 , 0]P (132) P (132) P (132)
= [0, 0, 0]I.
~ ~ ~ ~
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~ ~ ~ ~
~
e +
boundary( )e +
e +
The Positron and Its Linked Boundary
e+ = [1, 1, 1] 11 2 = [1, 1, 1][1/ 2, 1/ 2, 1/ 2]P 12 [1/ 2, 1/ 2, 1/ 2]P 23
= [1 , 1, 1][1/ 2, 1/ 2, 1/ 2][ 1/ 2, 1/ 2, 1/ 2]P 12 P 23 = [1, 2, 0]P 12 P 23 .
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~ ~
e+
~ ~ ~ ~
~ ~ ~ ~ ~ ~
u(3) d(3)
e - u(3)_
_
d(3)
e+
~ ~ ~ ~
~ ~ ~ ~
u(3)d(3)
e -u(3)_
_
d(3)
_
~ ~ ~ ~
~ ~ ~ ~
different
different
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Apparent Moral:There is topologicalpersistence in particle properties
for the surfaces.
To what extent do the surfacesrepresent the elementary particles?
To what extent does the framed braidand/or its algebraic representation
represent theelementary particles?
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Algebra Appendix
A permutation matrix P is a matrix whose columns are a permutation of the rows of the identity matrix. Such a matrix acts as a permutation on thestandard basis (column vectors that have a single unit entry). We shall refer to
P as a permutation. If D is a diagonal matrix and P is a permutation matrix,thenP D = D P P
where D P denotes the result of permuting the elements of D along the diagonalaccording to the permutation P.
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Let [a,b,c ] denote the matrix
[a,b,c ] =t a 0 00 t b 00 0 t c
.
Let
P 12 = 0 1 01 0 00 0 1
and
P 23=
1 0 0
0 0 10 1 0.
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1 =0 t 1 / 2 0
t 1 / 2 0 00 0 t 1 / 2
2 =t 1 / 2 0 0
0 0 t 1 / 2
0 t 1 / 2 0
This is the (framed) braid grouprepresentation with which we have
been working.