L11n189

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L11n188.gif

L11n188

L11n190.gif

L11n190

Contents

L11n189.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n189's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X17,6,18,1
Gauss code {1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, 5, -11, 7, -8, 6, -7, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n189 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^3-2 u^2 v^2+u^2 v+u^2-2 u v^3+5 u v^2-2 u v+v^4+v^3-2 v^2+v}{u v^2} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+4 q^{5/2}-\frac{3}{q^{5/2}}-5 q^{3/2}+\frac{4}{q^{3/2}}-\frac{1}{q^{11/2}}+6 \sqrt{q}-\frac{5}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z a^5+2 a^5 z^{-1} -z^3 a^3-5 z a^3-3 a^3 z^{-1} +2 z^3 a+3 z a+a z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -3 z a^{-1} +z^3 a^{-3} +z a^{-3} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -a^2 z^8-3 z^8 a^{-2} -4 z^8-a^5 z^7+4 a z^7-3 z^7 a^{-3} +6 a^2 z^6+11 z^6 a^{-2} -z^6 a^{-4} +18 z^6+7 a^5 z^5+3 a^3 z^5-4 a z^5+11 z^5 a^{-1} +11 z^5 a^{-3} +3 a^4 z^4-10 a^2 z^4-9 z^4 a^{-2} +3 z^4 a^{-4} -25 z^4-14 a^5 z^3-11 a^3 z^3-a z^3-12 z^3 a^{-1} -8 z^3 a^{-3} -7 a^4 z^2+3 z^2 a^{-2} -z^2 a^{-4} +11 z^2+10 a^5 z+10 a^3 z+a z+2 z a^{-1} +z a^{-3} +3 a^4+3 a^2+1-2 a^5 z^{-1} -3 a^3 z^{-1} -a 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 \
-6-5-4-3-2-1012345χ
10           1-1
8          2 2
6         21 -1
4        32  1
2       32   -1
0     133    -1
-2     34     1
-4   122      -1
-6    3       3
-8  11        0
-101           1
-121           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-6 {\mathbb Z} {\mathbb Z}
r=-5
r=-4 {\mathbb Z}
r=-3 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3} {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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L11n188

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L11n190