L11n232

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L11n231.gif

L11n231

L11n233.gif

L11n233

Contents

L11n232.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n232's Link Presentations]

Planar diagram presentation X10,1,11,2 X18,11,19,12 X8,9,1,10 X22,19,9,20 X20,6,21,5 X4,22,5,21 X7,15,8,14 X12,4,13,3 X13,16,14,17 X15,7,16,6 X2,18,3,17
Gauss code {1, -11, 8, -6, 5, 10, -7, -3}, {3, -1, 2, -8, -9, 7, -10, 9, 11, -2, 4, -5, 6, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n232 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 \left(-v^2\right)+2 u^3 v-u^3+u^2 v^2-u^2 v+u^2+u v^3-u v^2+u v-v^3+2 v^2-v}{u^{3/2} v^{3/2}} (db)
Jones polynomial 2 q^{9/2}-4 q^{7/2}+4 q^{5/2}-\frac{1}{q^{5/2}}-5 q^{3/2}+\frac{3}{q^{3/2}}-q^{11/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z^5 a^{-1} +a z^3-3 z^3 a^{-1} +2 z^3 a^{-3} +a z-2 z a^{-1} +4 z a^{-3} -z a^{-5} + a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial z^7 a^{-5} -5 z^5 a^{-5} +8 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -9 z^6 a^{-4} +10 z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} +z^9 a^{-3} -z^7 a^{-3} -11 z^5 a^{-3} +18 z^3 a^{-3} -8 z a^{-3} + a^{-3} z^{-1} +4 z^8 a^{-2} -17 z^6 a^{-2} +17 z^4 a^{-2} +a^2 z^2-4 z^2 a^{-2} +z^9 a^{-1} +a z^7-z^7 a^{-1} -4 a z^5-10 z^5 a^{-1} +5 a z^3+15 z^3 a^{-1} -2 a z-5 z a^{-1} +2 z^8-8 z^6+7 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 \
-2-10123456χ
12        11
10       1 -1
8      31 2
6     22  0
4    32   1
2  122    1
0  33     0
-2 12      1
-4 2       -2
-61        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3} {\mathbb Z}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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L11n231.gif

L11n231

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L11n233