L11a197

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L11a196

L11a198.gif

L11a198

Contents

L11a197.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a197's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X16,6,17,5 X18,12,19,11 X20,16,21,15 X12,18,13,17 X14,22,15,21 X2738 X4,14,5,13 X6,20,1,19
Gauss code {1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 5, -7, 10, -8, 6, -4, 7, -5, 11, -6, 8, -3}
A Braid Representative
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A Morse Link Presentation L11a197 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^4-t(2)^4+4 t(1)^2 t(2)^3-10 t(1) t(2)^3+5 t(2)^3-8 t(1)^2 t(2)^2+17 t(1) t(2)^2-8 t(2)^2+5 t(1)^2 t(2)-10 t(1) t(2)+4 t(2)-t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial q^{21/2}-4 q^{19/2}+9 q^{17/2}-15 q^{15/2}+21 q^{13/2}-24 q^{11/2}+24 q^{9/2}-22 q^{7/2}+15 q^{5/2}-10 q^{3/2}+4 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-3} -3 z^5 a^{-5} -z^5 a^{-7} +z^3 a^{-1} +2 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} +z^3 a^{-9} +5 z a^{-3} -6 z a^{-5} +2 z a^{-7} +2 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} (db)
Kauffman polynomial z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -9 z^5 a^{-11} +6 z^3 a^{-11} -z a^{-11} +7 z^8 a^{-10} -15 z^6 a^{-10} +10 z^4 a^{-10} -3 z^2 a^{-10} +6 z^9 a^{-9} -4 z^7 a^{-9} -12 z^5 a^{-9} +11 z^3 a^{-9} -3 z a^{-9} +2 z^{10} a^{-8} +15 z^8 a^{-8} -41 z^6 a^{-8} +29 z^4 a^{-8} -6 z^2 a^{-8} - a^{-8} +13 z^9 a^{-7} -13 z^7 a^{-7} -18 z^5 a^{-7} +25 z^3 a^{-7} -8 z a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +19 z^8 a^{-6} -46 z^6 a^{-6} +31 z^4 a^{-6} -2 z^2 a^{-6} -3 a^{-6} +7 z^9 a^{-5} +4 z^7 a^{-5} -31 z^5 a^{-5} +33 z^3 a^{-5} -13 z a^{-5} +3 a^{-5} z^{-1} +11 z^8 a^{-4} -17 z^6 a^{-4} +10 z^4 a^{-4} -3 a^{-4} +9 z^7 a^{-3} -15 z^5 a^{-3} +12 z^3 a^{-3} -7 z a^{-3} +2 a^{-3} z^{-1} +4 z^6 a^{-2} -4 z^4 a^{-2} +z^5 a^{-1} -z^3 a^{-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          3 3
18         61 -5
16        93  6
14       126   -6
12      129    3
10     1212     0
8    1012      -2
6   613       7
4  49        -5
2 17         6
0 3          -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a196

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L11a198