L11n190

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L11n189.gif

L11n189

L11n191.gif

L11n191

Contents

L11n190.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n190 at Knotilus!


Link Presentations

[edit Notes on L11n190's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,16,6,17 X20,15,21,16 X18,21,19,22 X14,19,15,20 X2738 X4,11,5,12 X17,6,18,1
Gauss code {1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, 5, -11, -7, 8, -6, 7, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n190 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-u^2 v^3+4 u^2 v^2-3 u^2 v+u^2+4 u v^3-7 u v^2+4 u v+v^4-3 v^3+4 v^2-v}{u v^2} (db)
Jones polynomial \frac{10}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{6}{q^{17/2}}-\frac{9}{q^{15/2}}+\frac{10}{q^{13/2}}-\frac{12}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -4 a^7 z^3-6 a^7 z-2 a^7 z^{-1} +a^5 z^5+a^5 z^3+a^5 z-2 a^3 z^3-2 a^3 z (db)
Kauffman polynomial -z^6 a^{12}+3 z^4 a^{12}-3 z^2 a^{12}+a^{12}-3 z^7 a^{11}+9 z^5 a^{11}-8 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -3 z^8 a^{10}+4 z^6 a^{10}+8 z^4 a^{10}-11 z^2 a^{10}+3 a^{10}-z^9 a^9-7 z^7 a^9+26 z^5 a^9-24 z^3 a^9+12 z a^9-3 a^9 z^{-1} -6 z^8 a^8+8 z^6 a^8+7 z^4 a^8-9 z^2 a^8+3 a^8-z^9 a^7-7 z^7 a^7+19 z^5 a^7-16 z^3 a^7+7 z a^7-2 a^7 z^{-1} -3 z^8 a^6+2 z^6 a^6-2 z^4 a^6+2 z^2 a^6-3 z^7 a^5+2 z^5 a^5-3 z^3 a^5-z^6 a^4-4 z^4 a^4+3 z^2 a^4-3 z^3 a^3+2 z a^3 (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
-2         22
-4        41-3
-6       41 3
-8      64  -2
-10     64   2
-12    46    2
-14   56     -1
-16  25      3
-18 14       -3
-20 2        2
-221         -1
Integral Khovanov Homology

(db, data source)

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

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L11n189.gif

L11n189

L11n191.gif

L11n191