L11a241

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L11a240.gif

L11a240

L11a242.gif

L11a242

Contents

L11a241.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a241's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X18,5,19,6 X20,16,21,15 X16,20,17,19 X14,22,15,21 X2738 X4,11,5,12 X6,17,1,18
Gauss code {1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, -7, 11, -5, 7, -6, 8, -3}
A Braid Representative
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A Morse Link Presentation L11a241 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^2-6 u^2 v+5 u^2-6 u v^2+13 u v-6 u+5 v^2-6 v+2}{u v} (db)
Jones polynomial q^{9/2}-\frac{6}{q^{9/2}}-4 q^{7/2}+\frac{9}{q^{7/2}}+7 q^{5/2}-\frac{14}{q^{5/2}}-11 q^{3/2}+\frac{16}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+15 \sqrt{q}-\frac{16}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} -3 z a^5-a^5 z^{-1} +3 z^3 a^3+z a^3-z^5 a+z^3 a+z a-z^5 a^{-1} -z^3 a^{-1} -2 z a^{-1} +z^3 a^{-3} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-3 a^3 z^9-7 a z^9-4 z^9 a^{-1} -3 a^4 z^8-7 a^2 z^8-6 z^8 a^{-2} -10 z^8-3 a^5 z^7+3 a^3 z^7+14 a z^7+4 z^7 a^{-1} -4 z^7 a^{-3} -2 a^6 z^6+20 a^2 z^6+17 z^6 a^{-2} -z^6 a^{-4} +36 z^6-a^7 z^5+3 a^5 z^5-7 a^3 z^5-11 a z^5+11 z^5 a^{-1} +11 z^5 a^{-3} +3 a^6 z^4+5 a^4 z^4-24 a^2 z^4-13 z^4 a^{-2} +2 z^4 a^{-4} -41 z^4+3 a^7 z^3+a^5 z^3+9 a^3 z^3+5 a z^3-12 z^3 a^{-1} -6 z^3 a^{-3} -3 a^4 z^2+10 a^2 z^2+4 z^2 a^{-2} +17 z^2-3 a^7 z-2 a^5 z-3 a^3 z-3 a z+z a^{-1} -a^6+a^7 z^{-1} +a^5 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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         41 -3
4        73  4
2       84   -4
0      87    1
-2     99     0
-4    57      -2
-6   49       5
-8  25        -3
-10  4         4
-1212          -1
-141           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z} {\mathbb Z}
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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L11a240

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