L11n181

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L11n180.gif

L11n180

L11n182.gif

L11n182

Contents

L11n181.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n181 at Knotilus!


Link Presentations

[edit Notes on L11n181's Link Presentations]

Planar diagram presentation X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X20,10,21,9 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22
Gauss code {1, -11, -2, 10, -3, 9}, {4, -1, 7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, -7, 11, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n181 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-u^2 v^2+u^2 v+u^2-u v^3+3 u v^2-u v+v^4+v^3-v^2}{u v^2} (db)
Jones polynomial -q^{7/2}+2 q^{5/2}-\frac{2}{q^{5/2}}-2 q^{3/2}+\frac{2}{q^{3/2}}-\frac{1}{q^{11/2}}+3 \sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z+2 a^5 z^{-1} -a^3 z^3-5 a^3 z-3 a^3 z^{-1} -z a^{-3} +a z^3+z^3 a^{-1} +a z+a z^{-1} +z a^{-1} (db)
Kauffman polynomial a^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+2 a^5 z^{-1} -2 a^4 z^4+6 a^4 z^2-3 a^4-2 a^3 z^5+z^5 a^{-3} +9 a^3 z^3-3 z^3 a^{-3} -9 a^3 z+z a^{-3} +3 a^3 z^{-1} +2 z^6 a^{-2} -7 z^4 a^{-2} +4 a^2 z^2+5 z^2 a^{-2} -3 a^2+z^7 a^{-1} +2 a z^5-2 z^5 a^{-1} -3 a z^3-z^3 a^{-1} +z a^{-1} +a z^{-1} +2 z^6-5 z^4+3 z^2-1 (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 \
-6-5-4-3-2-101234χ
8          11
6         1 -1
4        11 0
2       21  -1
0     121   0
-2     23    1
-4   121     0
-6    2      2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L11n180

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L11n182