L11n82

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L11n81.gif

L11n81

L11n83.gif

L11n83

Contents

L11n82.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n82's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X5,14,6,15 X3,10,4,11 X11,16,12,17 X15,12,16,13 X13,22,14,5 X18,9,19,10 X17,2,18,3 X8,19,9,20
Gauss code {1, 10, -5, -3}, {-4, -1, 2, -11, 9, 5, -6, 7, -8, 4, -7, 6, -10, -9, 11, -2, 3, 8}
A Braid Representative
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A Morse Link Presentation L11n82 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{4 u v^3-6 u v^2+4 u v-u-v^3+4 v^2-6 v+4}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{4}{q^{21/2}}+\frac{2}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +z^3 a^{11}+3 z a^{11}+2 a^{11} z^{-1} -z^5 a^9-2 z^3 a^9-z a^9-a^9 z^{-1} -2 z^5 a^7-5 z^3 a^7-z a^7+a^7 z^{-1} -z^5 a^5-3 z^3 a^5-3 z a^5-a^5 z^{-1} (db)
Kauffman polynomial -3 z^4 a^{14}+6 z^2 a^{14}-3 a^{14}-z^7 a^{13}-z^5 a^{13}+3 z^3 a^{13}-3 z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+5 z^6 a^{12}-16 z^4 a^{12}+20 z^2 a^{12}-7 a^{12}-z^9 a^{11}-7 z^5 a^{11}+14 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} -4 z^8 a^{10}+5 z^6 a^{10}-8 z^4 a^{10}+11 z^2 a^{10}-4 a^{10}-z^9 a^9-2 z^7 a^9-z^5 a^9+8 z^3 a^9-5 z a^9+a^9 z^{-1} -2 z^8 a^8-2 z^6 a^8+8 z^4 a^8-3 z^2 a^8-3 z^7 a^7+4 z^5 a^7-2 z a^7+a^7 z^{-1} -2 z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+3 z^3 a^5-3 z a^5+a^5 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 \
-9-8-7-6-5-4-3-2-10χ
-4         11
-6        21-1
-8       4  4
-10      42  -2
-12     64   2
-14    55    0
-16   55     0
-18  25      3
-20 25       -3
-22 2        2
-242         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-9 {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\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

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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L11n81.gif

L11n81

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L11n83