L11n231

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L11n230.gif

L11n230

L11n232.gif

L11n232

Contents

L11n231.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n231's Link Presentations]

Planar diagram presentation X10,1,11,2 X16,11,17,12 X5,21,6,20 X12,4,13,3 X14,8,15,7 X6,14,7,13 X17,9,18,22 X21,19,22,18 X8,9,1,10 X19,5,20,4 X2,16,3,15
Gauss code {1, -11, 4, 10, -3, -6, 5, -9}, {9, -1, 2, -4, 6, -5, 11, -2, -7, 8, -10, 3, -8, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
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A Morse Link Presentation L11n231 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 v^3-2 u^3 v^2+u^3 v-u^2 v^3+5 u^2 v^2-3 u^2 v-3 u v^2+5 u v-u+v^2-2 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{15/2}+3 q^{13/2}-6 q^{11/2}+8 q^{9/2}-9 q^{7/2}+8 q^{5/2}-8 q^{3/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-3} -z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} -3 z^3 a^{-1} +9 z^3 a^{-3} -3 z^3 a^{-5} -2 z a^{-1} +7 z a^{-3} -3 z a^{-5} + a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial -z^9 a^{-3} -z^9 a^{-5} -3 z^8 a^{-2} -5 z^8 a^{-4} -2 z^8 a^{-6} -3 z^7 a^{-1} -4 z^7 a^{-3} -2 z^7 a^{-5} -z^7 a^{-7} +7 z^6 a^{-2} +12 z^6 a^{-4} +4 z^6 a^{-6} -z^6+10 z^5 a^{-1} +20 z^5 a^{-3} +10 z^5 a^{-5} -6 z^4 a^{-4} -6 z^4 a^{-6} -3 z^4 a^{-8} +3 z^4-9 z^3 a^{-1} -20 z^3 a^{-3} -13 z^3 a^{-5} -3 z^3 a^{-7} -z^3 a^{-9} -2 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} -2 z^2+3 z a^{-1} +9 z a^{-3} +6 z a^{-5} +z a^{-7} +z a^{-9} + a^{-4} - a^{-3} z^{-1} - a^{-5} z^{-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 \
-3-2-10123456χ
16         11
14        2 -2
12       41 3
10      53  -2
8     43   1
6    45    1
4   44     0
2  25      3
0 13       -2
-2 2        2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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L11n230.gif

L11n230

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L11n232