L11a266

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L11a265.gif

L11a265

L11a267.gif

L11a267

Contents

L11a266.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a266's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X4,22,5,21 X14,6,15,5 X16,20,17,19 X18,8,19,7 X6,18,7,17 X20,16,21,15 X8,14,1,13
Gauss code {1, -4, 2, -5, 6, -9, 8, -11}, {4, -1, 3, -2, 11, -6, 10, -7, 9, -8, 7, -10, 5, -3}
A Braid Representative
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A Morse Link Presentation L11a266 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2)^2 t(1)^3-3 t(2) t(1)^3+t(1)^3+2 t(2)^3 t(1)^2-9 t(2)^2 t(1)^2+9 t(2) t(1)^2-3 t(1)^2-3 t(2)^3 t(1)+9 t(2)^2 t(1)-9 t(2) t(1)+2 t(1)+t(2)^3-3 t(2)^2+2 t(2)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial 18 q^{9/2}-16 q^{7/2}+11 q^{5/2}-7 q^{3/2}+q^{21/2}-4 q^{19/2}+8 q^{17/2}-12 q^{15/2}+16 q^{13/2}-19 q^{11/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-9} -z^5 a^{-7} +z a^{-7} -2 z^5 a^{-5} -3 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} -z^5 a^{-3} +2 z a^{-3} + a^{-3} z^{-1} +z^3 a^{-1} +z a^{-1} (db)
Kauffman polynomial z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -10 z^5 a^{-11} +6 z^3 a^{-11} -z a^{-11} +6 z^8 a^{-10} -14 z^6 a^{-10} +7 z^4 a^{-10} +4 z^9 a^{-9} -z^7 a^{-9} -15 z^5 a^{-9} +13 z^3 a^{-9} -3 z a^{-9} +z^{10} a^{-8} +11 z^8 a^{-8} -28 z^6 a^{-8} +17 z^4 a^{-8} -2 z^2 a^{-8} +7 z^9 a^{-7} -4 z^7 a^{-7} -16 z^5 a^{-7} +19 z^3 a^{-7} -5 z a^{-7} +z^{10} a^{-6} +10 z^8 a^{-6} -20 z^6 a^{-6} +12 z^4 a^{-6} -z^2 a^{-6} +3 z^9 a^{-5} +6 z^7 a^{-5} -20 z^5 a^{-5} +20 z^3 a^{-5} -8 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -4 z^6 a^{-4} -z^4 a^{-4} +2 z^2 a^{-4} - a^{-4} +5 z^7 a^{-3} -8 z^5 a^{-3} +6 z^3 a^{-3} -4 z a^{-3} + a^{-3} z^{-1} +3 z^6 a^{-2} -5 z^4 a^{-2} +2 z^2 a^{-2} +z^5 a^{-1} -2 z^3 a^{-1} +z 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         51 -4
16        73  4
14       95   -4
12      107    3
10     910     1
8    79      -2
6   49       5
4  37        -4
2 15         4
0 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11a267