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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 L11n321's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code {1, -4, -3, 7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, 4, -5, -7, 6, -8, 11, -10, 8}
A Braid Representative
A Morse Link Presentation L11n321 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^2 w^3+2 u v^2 w^2-u v^2 w+u v w^3-4 u v w^2+2 u v w+u w^2-u w+v^3 w^2-v^3 w-2 v^2 w^2+4 v^2 w-v^2+v w^2-2 v w+v}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial -2 q^3+5 q^2-6 q+9-8 q^{-1} +9 q^{-2} -6 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^2 z^6+a^4 z^4-5 a^2 z^4+3 z^4+3 a^4 z^2-11 a^2 z^2-2 z^2 a^{-2} +9 z^2+3 a^4-9 a^2-2 a^{-2} +8+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^6 z^6-4 a^6 z^4+4 a^6 z^2-a^6+2 a^5 z^7-7 a^5 z^5+6 a^5 z^3-a^5 z+2 a^4 z^8-5 a^4 z^6+2 a^4 z^4-2 a^4 z^2-a^4 z^{-2} +3 a^4+a^3 z^9+a^3 z^7-10 a^3 z^5+13 a^3 z^3+3 z^3 a^{-3} -8 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +5 a^2 z^8-16 a^2 z^6+z^6 a^{-2} +24 a^2 z^4+4 z^4 a^{-2} -23 a^2 z^2-7 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2+3 a^{-2} +a z^9+2 a z^7+3 z^7 a^{-1} -9 a z^5-6 z^5 a^{-1} +15 a z^3+11 z^3 a^{-1} -12 a z-7 z a^{-1} +2 a z^{-1} +3 z^8-9 z^6+22 z^4-24 z^2- z^{-2} +11 (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 \
7         2-2
5        3 3
3       32 -1
1      63  3
-1     56   1
-3    43    1
-5   25     3
-7  24      -2
-9 13       2
-11 1        -1
-131         1
Integral Khovanov Homology

(db, data source)

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

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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