L10n101

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L10n100.gif

L10n100

L10n102.gif

L10n102

Contents

L10n101.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n101's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X20,16,17,15 X16,20,9,19 X12,18,13,17 X2536 X9,1,10,4
Gauss code {1, -9, -2, 10}, {9, -1, -3, 4}, {8, -5, 7, -6}, {-10, 2, 5, -8, -4, 3, 6, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L10n101 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(4)-1) \left(-t(4)^2+t(1) t(4)+t(2) t(4)-t(1) t(3) t(4)+t(1) t(2) t(3) t(4)-t(2) t(3) t(4)-t(4)+t(1) t(2) t(3)\right)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} (db)
Jones polynomial 2 q^{9/2}-4 q^{7/2}+2 q^{5/2}-4 q^{3/2}-q^{17/2}-q^{13/2}-3 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial - a^{-9} z^{-3} - a^{-9} z^{-1} +3 a^{-7} z^{-3} +2 z a^{-7} +5 a^{-7} z^{-1} -3 a^{-5} z^{-3} -3 z a^{-5} -6 a^{-5} z^{-1} -z^5 a^{-3} -3 z^3 a^{-3} + a^{-3} z^{-3} -z a^{-3} + a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial z^7 a^{-9} -7 z^5 a^{-9} +15 z^3 a^{-9} - a^{-9} z^{-3} -14 z a^{-9} +6 a^{-9} z^{-1} +z^6 a^{-8} -9 z^4 a^{-8} +19 z^2 a^{-8} +3 a^{-8} z^{-2} -13 a^{-8} +z^7 a^{-7} -10 z^5 a^{-7} +28 z^3 a^{-7} -3 a^{-7} z^{-3} -27 z a^{-7} +14 a^{-7} z^{-1} +2 z^6 a^{-6} -14 z^4 a^{-6} +33 z^2 a^{-6} +6 a^{-6} z^{-2} -24 a^{-6} +z^7 a^{-5} -4 z^5 a^{-5} +10 z^3 a^{-5} -3 a^{-5} z^{-3} -16 z a^{-5} +12 a^{-5} z^{-1} +3 z^6 a^{-4} -10 z^4 a^{-4} +14 z^2 a^{-4} +3 a^{-4} z^{-2} -11 a^{-4} +z^7 a^{-3} -6 z^3 a^{-3} - a^{-3} z^{-3} +3 a^{-3} z^{-1} +2 z^6 a^{-2} -5 z^4 a^{-2} + a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +3 z a^{-1} - a^{-1} z^{-1} (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 \
-2-1012345678χ
18          11
16          11
14        1  1
12      4    4
10     351   1
8    2      2
6    31     2
4  42       2
2 13        2
0 1         -1
-21          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2 i=4
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{4}
r=1 {\mathbb Z}^{2}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z} {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}_2 {\mathbb Z}^{5} {\mathbb Z}^{4}
r=5 {\mathbb Z}
r=6 {\mathbb Z}_2 {\mathbb Z}
r=7
r=8 {\mathbb Z} {\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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L10n100.gif

L10n100

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L10n102