L11n343

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L11n342.gif

L11n342

L11n344.gif

L11n344

Contents

L11n343.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n343's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,19,12,18 X7,14,8,15 X13,8,14,9 X15,13,16,22 X17,21,18,20 X21,17,22,16 X19,5,20,12 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 9}, {-5, 4, -6, 8, -7, 3, -9, 7, -8, 6}
A Braid Representative
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A Morse Link Presentation L11n343 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(v-1) (w-1) \left(u v-2 u+2 v^2-v\right)}{\sqrt{u} v^{3/2} \sqrt{w}} (db)
Jones polynomial -2 q^3+5 q^2-6 q+8-8 q^{-1} +8 q^{-2} -5 q^{-3} +4 q^{-4} - q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+z^4 a^2+a^2 z^{-2} +2 z^4+4 z^2+3-2 z^2 a^{-2} - a^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8+a^5 z^7-2 a^3 z^7+3 z^7 a^{-1} +a^6 z^6-12 a^2 z^6+z^6 a^{-2} -10 z^6-2 a^5 z^5+4 a^3 z^5-6 z^5 a^{-1} -5 a^6 z^4-9 a^4 z^4+15 a^2 z^4+4 z^4 a^{-2} +23 z^4-3 a^5 z^3-10 a^3 z^3-a z^3+9 z^3 a^{-1} +3 z^3 a^{-3} +8 a^6 z^2+13 a^4 z^2-8 a^2 z^2-7 z^2 a^{-2} -20 z^2+6 a^5 z+8 a^3 z-4 z a^{-1} -2 z a^{-3} -5 a^6-8 a^4-a^2+2 a^{-2} +5-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^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 \
-6-5-4-3-2-10123χ
7         2-2
5        3 3
3       32 -1
1      53  2
-1     55   0
-3    33    0
-5   25     3
-7  23      -1
-9 14       3
-11          0
-131         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}_2^{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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L11n342

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L11n344