L11n286

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L11n285.gif

L11n285

L11n287.gif

L11n287

Contents

L11n286.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n286 at Knotilus!


Link Presentations

[edit Notes on L11n286's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X15,1,16,4 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code {1, -4, -3, 6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, 4, -5, -6, 10, -11, 7, -9, 8, -7, 11}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n286 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v w^2-3 u v w-2 u w^2+2 u w-2 v^2 w+2 v^2+3 v w-2 v}{\sqrt{u} v w} (db)
Jones polynomial - q^{-7} +2 q^{-6} -3 q^{-5} +5 q^{-4} -5 q^{-3} +2 q^2+6 q^{-2} -2 q-5 q^{-1} +5 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^6-a^6+z^4 a^4+2 z^2 a^4+2 a^4+z^4 a^2+z^2 a^2+a^2 z^{-2} +a^2-3 z^2-2 z^{-2} -4+ a^{-2} z^{-2} +2 a^{-2} (db)
Kauffman polynomial a^5 z^9+a^3 z^9+2 a^6 z^8+4 a^4 z^8+2 a^2 z^8+a^7 z^7-2 a^5 z^7-a^3 z^7+2 a z^7-10 a^6 z^6-17 a^4 z^6-5 a^2 z^6+2 z^6-5 a^7 z^5-5 a^5 z^5-4 a^3 z^5-3 a z^5+z^5 a^{-1} +15 a^6 z^4+23 a^4 z^4+3 a^2 z^4-5 z^4+7 a^7 z^3+10 a^5 z^3+2 a^3 z^3-2 a z^3-z^3 a^{-1} -8 a^6 z^2-16 a^4 z^2-a^2 z^2+3 z^2 a^{-2} +10 z^2-2 a^7 z-4 a^5 z+6 a z+4 z a^{-1} +2 a^6+4 a^4-2 a^2-4 a^{-2} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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 \
-7-6-5-4-3-2-1012χ
5         22
3        110
1       41 3
-1      33  0
-3     32   1
-5    34    1
-7   22     0
-9  13      2
-11 12       -1
-13 1        1
-151         -1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

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

See/edit the Link_Splice_Base (expert).

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L11n285.gif

L11n285

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L11n287