L11n107

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L11n106.gif

L11n106

L11n108.gif

L11n108

Contents

L11n107.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n107's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X13,19,14,18 X17,11,18,10 X21,9,22,8 X7,17,8,16 X9,21,10,20 X15,5,16,22 X19,15,20,14 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {10, -1, -6, 5, -7, 4, -11, 2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8}
A Braid Representative
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A Morse Link Presentation L11n107 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(3 t(2)^2-4 t(2)+3\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial 12 q^{9/2}-11 q^{7/2}+6 q^{5/2}-3 q^{3/2}+q^{21/2}-3 q^{19/2}+7 q^{17/2}-10 q^{15/2}+13 q^{13/2}-14 q^{11/2} (db)
Signature 3 (db)
HOMFLY-PT polynomial -2 z^5 a^{-5} -z^5 a^{-7} +3 z^3 a^{-3} -5 z^3 a^{-5} -z^3 a^{-7} +z^3 a^{-9} +5 z a^{-3} -5 z a^{-5} -z a^{-7} +z a^{-9} +2 a^{-3} z^{-1} -2 a^{-5} z^{-1} - a^{-7} z^{-1} + a^{-9} z^{-1} (db)
Kauffman polynomial z^6 a^{-12} -3 z^4 a^{-12} +3 z^2 a^{-12} - a^{-12} +3 z^7 a^{-11} -8 z^5 a^{-11} +6 z^3 a^{-11} -z a^{-11} +4 z^8 a^{-10} -8 z^6 a^{-10} +z^4 a^{-10} +2 z^2 a^{-10} +2 z^9 a^{-9} +4 z^7 a^{-9} -19 z^5 a^{-9} +12 z^3 a^{-9} -2 z a^{-9} - a^{-9} z^{-1} +10 z^8 a^{-8} -21 z^6 a^{-8} +11 z^4 a^{-8} -6 z^2 a^{-8} +3 a^{-8} +2 z^9 a^{-7} +8 z^7 a^{-7} -25 z^5 a^{-7} +19 z^3 a^{-7} -4 z a^{-7} - a^{-7} z^{-1} +6 z^8 a^{-6} -9 z^6 a^{-6} +7 z^4 a^{-6} -2 z^2 a^{-6} +7 z^7 a^{-5} -14 z^5 a^{-5} +19 z^3 a^{-5} -10 z a^{-5} +2 a^{-5} z^{-1} +3 z^6 a^{-4} +3 z^2 a^{-4} -3 a^{-4} +6 z^3 a^{-3} -7 z a^{-3} +2 a^{-3} z^{-1} (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 \
0123456789χ
22         1-1
20        2 2
18       51 -4
16      52  3
14     85   -3
12    65    1
10   68     2
8  56      -1
6 16       5
425        -3
23         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=0 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=9 {\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

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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L11n106.gif

L11n106

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L11n108