L11n216

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L11n215

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L11n217

Contents

L11n216.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n216's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,3,13,4 X18,8,19,7 X14,6,15,5 X17,9,18,22 X21,17,22,16 X20,13,21,14 X6,16,7,15 X4,20,5,19 X2,9,3,10 X8,11,1,12
Gauss code {1, -10, 2, -9, 4, -8, 3, -11}, {10, -1, 11, -2, 7, -4, 8, 6, -5, -3, 9, -7, -6, 5}
A Braid Representative
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A Morse Link Presentation L11n216 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2) t(1)^2+t(2)^2 t(1)-2 t(2) t(1)+t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{13/2}+3 q^{11/2}-5 q^{9/2}+7 q^{7/2}-8 q^{5/2}+8 q^{3/2}-8 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{1}{q^{5/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} -z a^{-5} +z^5 a^{-3} +2 z^3 a^{-3} +z a^{-3} +z^5 a^{-1} -a z^3+2 z^3 a^{-1} -a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -7 z^4 a^{-6} +2 z^2 a^{-6} +4 z^7 a^{-5} -10 z^5 a^{-5} +6 z^3 a^{-5} -2 z a^{-5} +3 z^8 a^{-4} -7 z^6 a^{-4} +6 z^4 a^{-4} -2 z^2 a^{-4} +z^9 a^{-3} +z^7 a^{-3} -6 z^5 a^{-3} +8 z^3 a^{-3} -2 z a^{-3} +4 z^8 a^{-2} -13 z^6 a^{-2} +a^2 z^4+20 z^4 a^{-2} -2 a^2 z^2-8 z^2 a^{-2} +z^9 a^{-1} -3 z^7 a^{-1} +3 a z^5+8 z^5 a^{-1} -6 a z^3-6 z^3 a^{-1} +2 a z+2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-3 z^6+8 z^4-6 z^2-1 (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 \
-3-2-10123456χ
14         11
12        2 -2
10       31 2
8      42  -2
6     43   1
4    44    0
2   44     0
0  26      4
-2 12       -1
-4 2        2
-61         -1
Integral Khovanov Homology

(db, data source)

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

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