L11n101

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L11n100.gif

L11n100

L11n102.gif

L11n102

Contents

L11n101.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n101's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,21,14,20 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, -3, 9, -8, -4, 7, -6, 5, 3, -9, 8}
A Braid Representative
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A Morse Link Presentation L11n101 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^3-5 u v^2+7 u v-3 u-3 v^3+7 v^2-5 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{2}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} +z^3 a^5-z a^5-2 a^5 z^{-1} -z^5 a^3-z^3 a^3+2 a^3 z^{-1} -z^5 a-z^3 a-z a-a z^{-1} +z^3 a^{-1} (db)
Kauffman polynomial 3 a^7 z^5-8 a^7 z^3+5 a^7 z-a^7 z^{-1} +a^6 z^8-4 a^6 z^4+3 a^6 z^2+a^5 z^9+2 a^5 z^5-11 a^5 z^3+9 a^5 z-2 a^5 z^{-1} +5 a^4 z^8-7 a^4 z^6-a^4 z^4+4 a^4 z^2-a^4+a^3 z^9+7 a^3 z^7-13 a^3 z^5+a^3 z^3+6 a^3 z-2 a^3 z^{-1} +4 a^2 z^8-6 a^2 z^4+z^4 a^{-2} +2 a^2 z^2+7 a z^7-8 a z^5+4 z^5 a^{-1} +a z^3-3 z^3 a^{-1} +2 a z-a z^{-1} +7 z^6-8 z^4+z^2 (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 \
-6-5-4-3-2-10123χ
6         1-1
4        3 3
2       41 -3
0      63  3
-2     65   -1
-4    65    1
-6   46     2
-8  36      -3
-10 25       3
-12 2        -2
-142         2
Integral Khovanov Homology

(db, data source)

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