L11n333

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L11n332.gif

L11n332

L11n334.gif

L11n334

Contents

L11n333.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n333 at Knotilus!


Link Presentations

[edit Notes on L11n333's Link Presentations]

Planar diagram presentation X6172 X11,16,12,17 X3849 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X19,1,20,4
Gauss code {1, -4, -3, 11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, -11, 8, -10, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n333 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v+w-1) (v w-v-w) (v-u w)}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial - q^{-7} +2 q^{-6} -3 q^{-5} +6 q^{-4} -5 q^{-3} +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+a^4 z^{-2} +3 a^4+z^4 a^2-2 a^2 z^{-2} -3 a^2-2 z^2+ z^{-2} + a^{-2} (db)
Kauffman polynomial a^5 z^9+a^3 z^9+2 a^6 z^8+5 a^4 z^8+3 a^2 z^8+a^7 z^7-a^5 z^7+a^3 z^7+3 a z^7-10 a^6 z^6-23 a^4 z^6-12 a^2 z^6+z^6-5 a^7 z^5-10 a^5 z^5-16 a^3 z^5-11 a z^5+16 a^6 z^4+36 a^4 z^4+19 a^2 z^4-z^4+7 a^7 z^3+17 a^5 z^3+23 a^3 z^3+15 a z^3+2 z^3 a^{-1} -11 a^6 z^2-31 a^4 z^2-20 a^2 z^2+z^2 a^{-2} +z^2-2 a^7 z-7 a^5 z-12 a^3 z-8 a z-z a^{-1} +3 a^6+11 a^4+11 a^2- a^{-2} +3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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         11
3        1 -1
1       41 3
-1      44  0
-3     221  1
-5    34    1
-7   32     1
-9  14      3
-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=-3 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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
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}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z} {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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L11n332

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L11n334