L11n184

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L11n183.gif

L11n183

L11n185.gif

L11n185

Contents

L11n184.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n184 at Knotilus!


Link Presentations

[edit Notes on L11n184's Link Presentations]

Planar diagram presentation X8192 X9,21,10,20 X6,21,1,22 X18,8,19,7 X3,10,4,11 X15,12,16,13 X5,14,6,15 X13,4,14,5 X11,16,12,17 X22,18,7,17 X19,2,20,3
Gauss code {1, 11, -5, 8, -7, -3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, -11, 2, 3, -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.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n184 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^4-t(2)^4-2 t(1) t(2)^3+2 t(2)^3+t(1) t(2)^2+2 t(1)^2 t(2)-2 t(1) t(2)-t(1)^2+2 t(1)}{t(1) t(2)^2} (db)
Jones polynomial -\frac{5}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{3}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{4}{q^{11/2}}+\sqrt{q}-\frac{2}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^7+2 z a^7-z^5 a^5-3 z^3 a^5-z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-6 z a^3-3 a^3 z^{-1} +z^3 a+2 z a+a z^{-1} (db)
Kauffman polynomial a^9 z^7-5 a^9 z^5+7 a^9 z^3-3 a^9 z+2 a^8 z^8-10 a^8 z^6+14 a^8 z^4-6 a^8 z^2+a^7 z^9-2 a^7 z^7-6 a^7 z^5+10 a^7 z^3-2 a^7 z+4 a^6 z^8-18 a^6 z^6+22 a^6 z^4-9 a^6 z^2+a^5 z^9-a^5 z^7-9 a^5 z^5+14 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-7 a^4 z^6+5 a^4 z^4+2 a^4 z^2-3 a^4+2 a^3 z^7-8 a^3 z^5+13 a^3 z^3-10 a^3 z+3 a^3 z^{-1} +a^2 z^6-3 a^2 z^4+6 a^2 z^2-3 a^2+2 a z^3-2 a z+a z^{-1} +z^2-1 (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 \
-8-7-6-5-4-3-2-101χ
2         1-1
0        1 1
-2       32 -1
-4      2   2
-6     23   1
-8    32    1
-10   12     1
-12  23      -1
-14 12       1
-16 1        -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2} {\mathbb Z}
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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L11n183

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L11n185