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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 L10a168's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,12,19,11 X20,16,17,15 X16,20,9,19 X12,18,13,17 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 3, -4}, {8, -5, 7, -6}, {10, -2, 5, -8, 4, -3, 6, -7}
A Braid Representative
A Morse Link Presentation L10a168 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(x-1) \left(u v w x-u v w-2 u v x-2 u w x+u w-u x^2+2 u x-2 v w x+v w-v x^2+2 v x+2 w x+x^2-x\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} (db)
Jones polynomial -\frac{6}{q^{9/2}}-q^{7/2}+\frac{9}{q^{7/2}}+3 q^{5/2}-\frac{14}{q^{5/2}}-8 q^{3/2}+\frac{12}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+10 \sqrt{q}-\frac{14}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} +a^5 z^{-3} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3-3 a^3 z^{-3} -4 a^3 z^{-1} -z a^{-3} -a z^5+a z^3+3 a z^{-3} +2 z^3 a^{-1} - a^{-1} z^{-3} +5 a z+7 a z^{-1} -z a^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-3 a^4 z^8-7 a^2 z^8-4 z^8-3 a^5 z^7-10 a^3 z^7-13 a z^7-6 z^7 a^{-1} -2 a^6 z^6+3 a^2 z^6-3 z^6 a^{-2} -2 z^6-a^7 z^5+3 a^5 z^5+27 a^3 z^5+37 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^6 z^4+8 a^4 z^4+21 a^2 z^4+4 z^4 a^{-2} +20 z^4+3 a^7 z^3-a^5 z^3-32 a^3 z^3-45 a z^3-15 z^3 a^{-1} +2 z^3 a^{-3} -15 a^4 z^2-36 a^2 z^2-z^2 a^{-2} -22 z^2-3 a^7 z+3 a^5 z+24 a^3 z+33 a z+14 z a^{-1} -z a^{-3} -a^6+11 a^4+24 a^2+13+a^7 z^{-1} -3 a^5 z^{-1} -12 a^3 z^{-1} -14 a z^{-1} -6 a^{-1} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-3} (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 \
8          11
6         31-2
4        5  5
2       53  -2
0      95   4
-2     79    2
-4    75     2
-6   27      5
-8  47       -3
-10 15        4
-12 1         -1
-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}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z} {\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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