L. H. Kauffman- Particle Topology, Braids and Braided Belts

8/3/2019 L. H. Kauffman- Particle Topology, Braids and Braided Belts

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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.

L. H. Kauffman- Particle Topology, Braids and Braided Belts

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