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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 L11n326's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X4,17,1,18
Gauss code {1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, -4, -5, 11, 6, -8, 7, -10, 9}
A Braid Representative
A Morse Link Presentation L11n326 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^3 w^3+u v^3 w^2+u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^2-v^2 w-v w^2+2 v w-v-w+1}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial - q^{-6} +2 q^{-5} -2 q^{-4} +q^3+5 q^{-3} -2 q^2-4 q^{-2} +3 q+5 q^{-1} -3 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^2 z^8+a^4 z^6-7 a^2 z^6+z^6+6 a^4 z^4-17 a^2 z^4+5 z^4-a^6 z^2+11 a^4 z^2-18 a^2 z^2+7 z^2-2 a^6+7 a^4-9 a^2+4+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+2 a^4 z^8+4 a^2 z^8+2 z^8+a^5 z^7-3 a^3 z^7-2 a z^7+2 z^7 a^{-1} -12 a^4 z^6-21 a^2 z^6+z^6 a^{-2} -8 z^6-5 a^5 z^5-3 a^3 z^5-6 a z^5-8 z^5 a^{-1} +2 a^6 z^4+28 a^4 z^4+37 a^2 z^4-4 z^4 a^{-2} +7 z^4+a^7 z^3+11 a^5 z^3+17 a^3 z^3+13 a z^3+6 z^3 a^{-1} -5 a^6 z^2-27 a^4 z^2-28 a^2 z^2+3 z^2 a^{-2} -3 z^2-2 a^7 z-7 a^5 z-12 a^3 z-8 a z-z a^{-1} +3 a^6+11 a^4+11 a^2- a^{-2} +3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2} (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         11
5        1 -1
3       21 1
1     121  0
-1    152   2
-3    34    1
-5   32     1
-7 113      3
-9 22       0
-11 1        1
-131         -1
Integral Khovanov Homology

(db, data source)

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