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

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Link Presentations

[edit Notes on L11a229's Link Presentations]

Planar diagram presentation X8192 X2,11,3,12 X12,3,13,4 X16,5,17,6 X6718 X4,15,5,16 X20,14,21,13 X18,10,19,9 X10,20,11,19 X22,18,7,17 X14,22,15,21
Gauss code {1, -2, 3, -6, 4, -5}, {5, -1, 8, -9, 2, -3, 7, -11, 6, -4, 10, -8, 9, -7, 11, -10}
A Braid Representative
A Morse Link Presentation L11a229 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^6-3 u^2 v^5+5 u^2 v^4-5 u^2 v^3+3 u^2 v^2+3 u v^5-7 u v^4+9 u v^3-7 u v^2+3 u v+3 v^4-5 v^3+5 v^2-3 v+1}{u v^3} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+8 q^{5/2}-13 q^{3/2}+17 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{18}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a z^9+a^3 z^7-7 a z^7+z^7 a^{-1} +5 a^3 z^5-20 a z^5+5 z^5 a^{-1} +10 a^3 z^3-29 a z^3+10 z^3 a^{-1} +9 a^3 z-21 a z+9 z a^{-1} +3 a^3 z^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-4 a^6 z^4+a^6 z^2+6 a^5 z^7-10 a^5 z^5+8 a^5 z^3-3 a^5 z+7 a^4 z^8-9 a^4 z^6+z^6 a^{-4} +4 a^4 z^4-3 z^4 a^{-4} +3 z^2 a^{-4} - a^{-4} +6 a^3 z^9-8 a^3 z^7+3 z^7 a^{-3} +12 a^3 z^5-7 z^5 a^{-3} -17 a^3 z^3+4 z^3 a^{-3} +12 a^3 z-3 a^3 z^{-1} +2 a^2 z^{10}+9 a^2 z^8+5 z^8 a^{-2} -25 a^2 z^6-10 z^6 a^{-2} +28 a^2 z^4+4 z^4 a^{-2} -18 a^2 z^2+5 a^2+11 a z^9+5 z^9 a^{-1} -25 a z^7-8 z^7 a^{-1} +35 a z^5+5 z^5 a^{-1} -41 a z^3-10 z^3 a^{-1} +25 a z+9 z a^{-1} -5 a z^{-1} -2 a^{-1} z^{-1} +2 z^{10}+7 z^8-24 z^6+27 z^4-20 z^2+5 (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 \
10           1-1
8          2 2
6         61 -5
4        72  5
2       106   -4
0      117    4
-2     1011     1
-4    810      -2
-6   510       5
-8  38        -5
-10  5         5
-1213          -2
-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}^{3}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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See/edit the Link Page master template (intermediate).

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