L11n267

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L11n266.gif

L11n266

L11n268.gif

L11n268

Contents

L11n267.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n267's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,19,12,18 X15,21,16,20 X17,9,18,22 X21,17,22,16 X19,13,20,12 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 9, 4, -3, -6, 8, -7, 5, -9, 6, -8, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n267 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v w^3-u v w^2+u v+u w^5-u w^4+u w-u+v w^5-v w^4+v w-v-w^5+w^3-w^2}{\sqrt{u} \sqrt{v} w^{5/2}} (db)
Jones polynomial -q^5+q^4-q^3-q^2+q-1+4 q^{-1} -2 q^{-2} +4 q^{-3} - q^{-4} + q^{-5} (db)
Signature 2 (db)
HOMFLY-PT polynomial 2 z^6-3 a^2 z^4-z^4 a^{-2} +12 z^4+a^4 z^2-14 a^2 z^2-6 z^2 a^{-2} -z^2 a^{-4} +23 z^2+4 a^4-18 a^2-7 a^{-2} +21+3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} +7 z^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+a^4 z^8+5 a^2 z^8+4 z^8-4 a^3 z^7-a z^7+3 z^7 a^{-1} -7 a^4 z^6-31 a^2 z^6-z^6 a^{-2} +z^6 a^{-4} -26 z^6-2 a^3 z^5-21 a z^5-20 z^5 a^{-1} +z^5 a^{-5} +18 a^4 z^4+61 a^2 z^4+3 z^4 a^{-2} -4 z^4 a^{-4} +50 z^4+21 a^3 z^3+53 a z^3+33 z^3 a^{-1} -3 z^3 a^{-3} -4 z^3 a^{-5} -22 a^4 z^2-54 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -42 z^2-24 a^3 z-45 a z-21 z a^{-1} +3 z a^{-3} +3 z a^{-5} +13 a^4+28 a^2+7 a^{-2} + a^{-4} +22+8 a^3 z^{-1} +15 a z^{-1} +7 a^{-1} z^{-1} - a^{-3} z^{-1} - a^{-5} z^{-1} -3 a^4 z^{-2} -8 a^2 z^{-2} -2 a^{-2} z^{-2} -7 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 \
-6-5-4-3-2-1012345χ
11           1-1
9            0
7         11 0
5       31   -2
3      121   0
1     341    0
-1    113     3
-3   13       2
-5  31        2
-7 14         3
-9            0
-111           1
Integral Khovanov Homology

(db, data source)

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