L10n37

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L10n36

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L10n38

Contents

L10n37.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n37's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,17,14,16 X9,15,10,14 X15,11,16,10 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {9, -1, -7, 8, -4, 5, 10, -2, -3, 4, -5, 3, -6, 7, -8, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n37 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(v^2-v+1\right) \left(-u v^2+u v+u+v^3+v^2-v\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -2 q^{9/2}+3 q^{7/2}-2 q^{5/2}-\frac{1}{q^{5/2}}+2 q^{3/2}-q^{13/2}+2 q^{11/2}-2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^5 a^{-1} +z^5 a^{-3} +a z^3-7 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +4 a z-13 z a^{-1} +9 z a^{-3} -2 z a^{-5} +4 a z^{-1} -8 a^{-1} z^{-1} +5 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +2 z^6 a^{-6} -7 z^4 a^{-6} +5 z^2 a^{-6} -2 a^{-6} +z^7 a^{-5} -2 z^5 a^{-5} -2 z^3 a^{-5} +z a^{-5} + a^{-5} z^{-1} +3 z^6 a^{-4} -13 z^4 a^{-4} +18 z^2 a^{-4} -9 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +9 z^3 a^{-3} -8 z a^{-3} +5 a^{-3} z^{-1} +2 z^6 a^{-2} -14 z^4 a^{-2} +28 z^2 a^{-2} -14 a^{-2} +a z^7+z^7 a^{-1} -7 a z^5-9 z^5 a^{-1} +15 a z^3+23 z^3 a^{-1} -13 a z-21 z a^{-1} +4 a z^{-1} +8 a^{-1} z^{-1} +z^6-8 z^4+15 z^2-8 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-4-3-2-10123456χ
14          11
12         1 -1
10        11 0
8       21  -1
6     111   -1
4     22    0
2   121     0
0    3      3
-2  1        1
-41          1
-61          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2 i=4
r=-4 {\mathbb Z} {\mathbb Z}
r=-3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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L10n36

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L10n38