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K
Eric Moorhouse
Department of Mathematics
K Eric Moorhouse Department of Mathematics Thomson modeled atoms - - PowerPoint PPT Presentation
K Eric Moorhouse Department of Mathematics Thomson modeled atoms - - PowerPoint PPT Presentation
K Eric Moorhouse Department of Mathematics Thomson modeled atoms as knots in the ether William Thomson (Lord Kelvin) 18241907 Bohrs model of the atom Niels Bohr 18851962 I R EA L L Y T H IN K K N O T S A R E A S G O O D A
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Thomson modeled atoms as knots in the ether
William Thomson (Lord Kelvin) 1824–1907
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Bohr’s model
- f the atom
Niels Bohr 1885–1962
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I R EA L L Y T H IN K K N O T S A R E A S G O O D A D ES CR IPT IO N O F A T O M S A S I’V E S EEN !
James Clerk Maxwell 1831–1879
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Began to make a catalog of knots… Peter Guthrie Tait 1831–1901
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Catalog of Knots
51 61 62 63 71 31 41 52 01 72 73 74 75 76 77 81 82
etc…
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Unknot 01
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=
Unknot 01
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The trefoil knot 31 is 3-colorable
Arcs of the knot diagram are colored red, green and blue. All three colors must be used.
- At each crossing point,
either one color or all three colors appear.
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The knot 41 is not 3-colorable
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The knot 41 is not 3-colorable
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The knot 41 is not 3-colorable
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The knot 41 is not 3-colorable
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The knot 41 is not 3-colorable
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The knot 41 is not 3-colorable
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The knot 41 is not 3-colorable
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This shows that
3-colorable not 3-colorable
≠
Trefoil knot 31 41
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And that
3-colorable not 3-colorable
≠
Trefoil knot 31 Unknot 01
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But why …
not 3-colorable not 3-colorable
≠
41
Unknot 01
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James Alexander 1888–1971
1–3x+x2
Alexander polynomial
- f the knot 41
41
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a b c d
C A D B
1–x x x –1 –1 1–x –1 1–x –1 x 1–x x
c a b d
A B C D
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1–3x+x2
Alexander polynomial
- f the knot 41
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Alexander Polynomials of Knots
etc…
52 31 41 51 01 1–x+x2 1–3x+x2 1–x+x2–x3+x4 2–3x+2x2 1 2–5x+2x2 1–3x+3x2 –3x3+x4 1–3x+5x2 –3x3+x4 1–x+x2–x3 +x4–x5+x6 61 63 62 71
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Prime Factorization of Knots
and their Alexander polynomials
(1–x+x2)(1–3x+x2) 41
1–3x+x2
1–x+x2
=
# 31 31 # 41
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But inequivalent knots sometimes have the same Alexander polynomial!
1–5x+12x2–15x3
+12x4–5x5+x6
1–5x+12x2–15x3
+12x4–5x5+x6
929 928
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Found connections between von Neumann algebras and geometric topology, resulting in a new polynomial invariant for knots. Vaughn Jones (1952– )
Awar ded t he Fi el ds M edal i n 1990
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Jones Polynomials of Knots
etc…
52 31 41 51 01
–x-4+x-3+x-1 x-2–x-1+1 –x+x2 –x-7+x-6–x-5 +x-4+x-2 –x-6+x-5–x-4 +2x-3–x-2+x-1
1 61 63 62 71 x-4–x-3+x-2
–2x-1+2–x+x2
x-5–2x-4+2x-3
–2x-2+2x –1+x –x-3+2x-2 –2x-1+3 –2x+2x2–x3 –x-10+x-9 –x-8+x-7 –x-6+x-5+x3
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These inequivalent knots have different Jones polynomials!
929
–x-6+3x-5–6x-4 +8x-3–8x-2+9x -1–7 +5x–3x2+x3 –x-2+3x-1–5 +8x–8x2+9x 3–8x 4 +5x 5–3x6+x7
928
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There do exist inequivalent knots with the same Jones polynomial. It is not known whether the unknot is the only knot with Jones polynomial = 1.
Unknot 01
Jones polynomial = 1
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Knots Links
vs.
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UW conducts intensive research into knots using the latest technology…
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www.pims.math.ca/KnotPlot/
Knotplot.lnk