L11n229

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L11n228.gif

L11n228

L11n230.gif

L11n230

Contents

L11n229.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n229 at Knotilus!


Link Presentations

[edit Notes on L11n229's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X7,14,8,15 X13,16,14,17 X15,8,16,1 X22,17,9,18 X18,21,19,22 X6,9,7,10 X4,19,5,20
Gauss code {1, -2, 3, -11, 4, -10, -5, 7}, {10, -1, 2, -3, -6, 5, -7, 6, 8, -9, 11, -4, 9, -8}
A Braid Representative
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A Morse Link Presentation L11n229 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(2 u v-u-v+2) \left(u^2 v^2+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{2}{q^{23/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -z a^{13}+z^5 a^{11}+5 z^3 a^{11}+5 z a^{11}-z^7 a^9-5 z^5 a^9-6 z^3 a^9-z a^9+a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-7 z a^7-a^7 z^{-1} (db)
Kauffman polynomial 2 a^{15} z+a^{14} z^4+a^{13} z^7-3 a^{13} z^5+2 a^{13} z^3+a^{13} z+2 a^{12} z^8-10 a^{12} z^6+17 a^{12} z^4-11 a^{12} z^2+a^{11} z^9-4 a^{11} z^7+6 a^{11} z^5-9 a^{11} z^3+5 a^{11} z+3 a^{10} z^8-14 a^{10} z^6+18 a^{10} z^4-8 a^{10} z^2+a^9 z^9-4 a^9 z^7+3 a^9 z^5-a^9 z+a^9 z^{-1} +a^8 z^8-4 a^8 z^6+2 a^8 z^4+3 a^8 z^2-a^8+a^7 z^7-6 a^7 z^5+11 a^7 z^3-7 a^7 z+a^7 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 \
-8-7-6-5-4-3-2-10χ
-6        11
-8       110
-10      2  2
-12     11  0
-14    32   1
-16   11    0
-18  23     -1
-20 11      0
-2212       -1
-242        2
Integral Khovanov Homology

(db, data source)

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

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

See/edit the Link_Splice_Base (expert).

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L11n228.gif

L11n228

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L11n230