L11n142

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L11n141.gif

L11n141

L11n143.gif

L11n143

Contents

L11n142.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n142 at Knotilus!


Link Presentations

[edit Notes on L11n142's Link Presentations]

Planar diagram presentation X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
Gauss code {1, 4, -3, -11, 5, -6}, {6, -1, -7, 3, 2, -5, 9, -10, 8, 7, -4, -2, 11, -8, 10, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n142 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^2-2 u^2 v-3 u v^2+7 u v-3 u-2 v+2}{u v} (db)
Jones polynomial q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^5+2 z a^5+a^5 z^{-1} -z^5 a^3-3 z^3 a^3-4 z a^3-2 a^3 z^{-1} -z^5 a-2 z^3 a+2 a z^{-1} +z^3 a^{-1} +z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-5 a^6 z^4+a^6 z^2+3 a^5 z^7-9 a^5 z^5+8 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+16 a^4 z^4-8 a^4 z^2+a^3 z^9-8 a^3 z^5+18 a^3 z^3-10 a^3 z+2 a^3 z^{-1} +4 a^2 z^8-16 a^2 z^6+27 a^2 z^4+z^4 a^{-2} -13 a^2 z^2-2 z^2 a^{-2} +a^2+a z^9-3 a z^7+5 a z^5+3 z^5 a^{-1} +a z^3-6 z^3 a^{-1} -4 a z+z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^8-3 z^6+7 z^4-6 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        2 2
2       21 -1
0      52  3
-2     33   0
-4    44    0
-6   34     1
-8  13      -2
-10 13       2
-12 1        -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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

See/edit the Link_Splice_Base (expert).

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L11n141

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L11n143