L11n177

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L11n176.gif

L11n176

L11n178.gif

L11n178

Contents

L11n177.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n177's Link Presentations]

Planar diagram presentation X8192 X20,9,21,10 X5,15,6,14 X18,12,19,11 X10,4,11,3 X7,13,8,12 X13,17,14,16 X17,7,18,22 X15,1,16,6 X4,21,5,22 X2,20,3,19
Gauss code {1, -11, 5, -10, -3, 9}, {-6, -1, 2, -5, 4, 6, -7, 3, -9, 7, -8, -4, 11, -2, 10, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n177 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-2 u^2 v^3+3 u^2 v^2-3 u^2 v+u^2-2 u v^4+5 u v^3-5 u v^2+5 u v-2 u+v^4-3 v^3+3 v^2-2 v+1}{u v^2} (db)
Jones polynomial -2 q^{15/2}+6 q^{13/2}-9 q^{11/2}+12 q^{9/2}-14 q^{7/2}+12 q^{5/2}-11 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z a^{-7} -z^5 a^{-5} -z^3 a^{-5} +z a^{-5} +z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} +z a^{-3} - a^{-3} z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial 3 z^3 a^{-9} -z a^{-9} +z^6 a^{-8} +6 z^4 a^{-8} -5 z^2 a^{-8} +4 z^7 a^{-7} -3 z^5 a^{-7} +4 z^3 a^{-7} -z a^{-7} +5 z^8 a^{-6} -8 z^6 a^{-6} +10 z^4 a^{-6} -6 z^2 a^{-6} +2 z^9 a^{-5} +6 z^7 a^{-5} -19 z^5 a^{-5} +12 z^3 a^{-5} -z a^{-5} +10 z^8 a^{-4} -21 z^6 a^{-4} +8 z^4 a^{-4} +2 z^9 a^{-3} +6 z^7 a^{-3} -27 z^5 a^{-3} +19 z^3 a^{-3} -2 z a^{-3} - a^{-3} z^{-1} +5 z^8 a^{-2} -11 z^6 a^{-2} +2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -11 z^5 a^{-1} +8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4+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 \
-3-2-10123456χ
16         22
14        4 -4
12       52 3
10      74  -3
8     75   2
6    68    2
4   56     -1
2  37      4
0 14       -3
-2 3        3
-41         -1
Integral Khovanov Homology

(db, data source)

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

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

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L11n176.gif

L11n176

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L11n178