L11a194

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L11a193.gif

L11a193

L11a195.gif

L11a195

Contents

L11a194.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a194's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X16,6,17,5 X20,14,21,13 X18,16,19,15 X14,20,15,19 X12,22,13,21 X2738 X4,12,5,11 X6,18,1,17
Gauss code {1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 10, -8, 5, -7, 6, -4, 11, -6, 7, -5, 8, -3}
A Braid Representative
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A Morse Link Presentation L11a194 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 u^2 v^2-5 u^2 v+2 u^2-5 u v^2+9 u v-5 u+2 v^2-5 v+3}{u v} (db)
Jones polynomial q^{21/2}-3 q^{19/2}+5 q^{17/2}-8 q^{15/2}+11 q^{13/2}-12 q^{11/2}+12 q^{9/2}-11 q^{7/2}+7 q^{5/2}-5 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-9} +z a^{-9} -z^5 a^{-7} -2 z^3 a^{-7} -2 z a^{-7} -z^5 a^{-5} +2 z a^{-5} -z^5 a^{-3} -2 z^3 a^{-3} -2 z a^{-3} - a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -5 z^9 a^{-7} -3 z^9 a^{-9} -2 z^8 a^{-4} -z^8 a^{-6} -3 z^8 a^{-8} -4 z^8 a^{-10} -2 z^7 a^{-3} +3 z^7 a^{-5} +15 z^7 a^{-7} +7 z^7 a^{-9} -3 z^7 a^{-11} -2 z^6 a^{-2} +5 z^6 a^{-6} +17 z^6 a^{-8} +13 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} -7 z^5 a^{-5} -19 z^5 a^{-7} -z^5 a^{-9} +10 z^5 a^{-11} +4 z^4 a^{-2} +3 z^4 a^{-4} -10 z^4 a^{-6} -23 z^4 a^{-8} -11 z^4 a^{-10} +3 z^4 a^{-12} +3 z^3 a^{-1} +7 z^3 a^{-3} +8 z^3 a^{-5} +7 z^3 a^{-7} -4 z^3 a^{-9} -7 z^3 a^{-11} -z^2 a^{-2} +5 z^2 a^{-6} +9 z^2 a^{-8} +4 z^2 a^{-10} -z^2 a^{-12} -3 z a^{-1} -5 z a^{-3} -3 z a^{-5} -z a^{-7} +z a^{-9} +z a^{-11} - a^{-2} + a^{-1} z^{-1} + 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 \
-2-10123456789χ
22           1-1
20          2 2
18         31 -2
16        52  3
14       63   -3
12      65    1
10     66     0
8    56      -1
6   37       4
4  24        -2
2 14         3
0 1          -1
-21           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a193.gif

L11a193

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L11a195