L11n161

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L11n160

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L11n162

Contents

L11n161.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n161's Link Presentations]

Planar diagram presentation X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X18,11,19,12 X5,13,6,12 X6,21,1,22 X13,20,14,21 X22,17,7,18 X19,4,20,5
Gauss code {1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, 6, 7, -9, -5, 4, -2, 10, -6, -11, 9, 8, -10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n161 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)+t(2)^2-3 t(2)+2}{t(1) t(2)} (db)
Jones polynomial -\frac{7}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{6}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^3+a^7 z-a^7 z^{-1} -a^5 z^5-2 a^5 z^3+a^5 z+3 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-4 a^3 z-2 a^3 z^{-1} +a z^3+a z (db)
Kauffman polynomial a^9 z^7-4 a^9 z^5+4 a^9 z^3-a^9 z+3 a^8 z^8-14 a^8 z^6+18 a^8 z^4-6 a^8 z^2-a^8+2 a^7 z^9-6 a^7 z^7-a^7 z^5+7 a^7 z^3-2 a^7 z+a^7 z^{-1} +7 a^6 z^8-30 a^6 z^6+34 a^6 z^4-9 a^6 z^2-3 a^6+2 a^5 z^9-4 a^5 z^7-6 a^5 z^5+10 a^5 z^3-5 a^5 z+3 a^5 z^{-1} +4 a^4 z^8-15 a^4 z^6+16 a^4 z^4-4 a^4 z^2-3 a^4+3 a^3 z^7-9 a^3 z^5+10 a^3 z^3-6 a^3 z+2 a^3 z^{-1} +a^2 z^6+3 a z^3-2 a z+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 \
-8-7-6-5-4-3-2-101χ
2         1-1
0        2 2
-2       32 -1
-4      41  3
-6     34   1
-8    43    1
-10   23     1
-12  24      -2
-14 12       1
-16 2        -2
-181         1
Integral Khovanov Homology

(db, data source)

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

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