L11n194

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L11n193.gif

L11n193

L11n195.gif

L11n195

Contents

L11n194.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n194 at Knotilus!


Link Presentations

[edit Notes on L11n194's Link Presentations]

Planar diagram presentation X8192 X16,6,17,5 X9,20,10,21 X21,10,22,11 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17
Gauss code {1, -7, 8, -9, 2, -11}, {9, -1, -3, 4, 7, -8, 10, -6, 5, -2, 11, -5, 6, 3, -4, -10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n194 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2-4 t(2)^2 t(1)+9 t(2) t(1)-4 t(1)+t(2)^2-4 t(2)+3}{t(1) t(2)} (db)
Jones polynomial 2 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z-a^3 z^5-a^3 z^3+a^3 z+a^3 z^{-1} -2 a z^5-6 a z^3+2 z^3 a^{-1} -6 a z-a z^{-1} +3 z a^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-3 a^4 z^8-5 a^2 z^8-2 z^8-4 a^5 z^7-5 a^3 z^7-2 a z^7-z^7 a^{-1} -3 a^6 z^6+a^4 z^6+6 a^2 z^6+2 z^6-a^7 z^5+6 a^5 z^5+8 a^3 z^5-2 a z^5-3 z^5 a^{-1} +6 a^6 z^4+4 a^4 z^4-4 a^2 z^4-3 z^4 a^{-2} -5 z^4+2 a^7 z^3-a^5 z^3+11 a z^3+8 z^3 a^{-1} -3 a^6 z^2-a^4 z^2+3 a^2 z^2+4 z^2 a^{-2} +5 z^2-a^7 z-3 a^3 z-9 a z-5 z a^{-1} -a^2+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 \
-6-5-4-3-2-10123χ
6         2-2
4        3 3
2       42 -2
0      73  4
-2     55   0
-4    66    0
-6   46     2
-8  25      -3
-10 14       3
-12 2        -2
-141         1
Integral Khovanov Homology

(db, data source)

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

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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L11n193

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L11n195