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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 L10a84's Link Presentations]

Planar diagram presentation X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X20,11,7,12 X10,19,11,20 X2,14,3,13 X12,4,13,3
Gauss code {1, -9, 10, -3, 4, -6}, {3, -1, 2, -8, 7, -10, 9, -2, 5, -4, 6, -5, 8, -7}
A Braid Representative
A Morse Link Presentation L10a84 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-2 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-2 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{7}{q^{7/2}}+7 q^{5/2}-\frac{11}{q^{5/2}}-11 q^{3/2}+\frac{13}{q^{3/2}}+\frac{1}{q^{11/2}}+13 \sqrt{q}-\frac{15}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+5 a z^5-2 z^5 a^{-1} +11 a z^3-6 z^3 a^{-1} +10 a z-6 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4-a^6 z^2+3 a^5 z^5-2 a^5 z^3+6 a^4 z^6+z^6 a^{-4} -7 a^4 z^4-2 z^4 a^{-4} +5 a^4 z^2+z^2 a^{-4} -a^4+8 a^3 z^7+4 z^7 a^{-3} -13 a^3 z^5-11 z^5 a^{-3} +13 a^3 z^3+8 z^3 a^{-3} -6 a^3 z-z a^{-3} +a^3 z^{-1} +6 a^2 z^8+5 z^8 a^{-2} -4 a^2 z^6-11 z^6 a^{-2} -7 a^2 z^4+3 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} -3 a^2+2 a z^9+2 z^9 a^{-1} +11 a z^7+7 z^7 a^{-1} -35 a z^5-30 z^5 a^{-1} +33 a z^3+26 z^3 a^{-1} -15 a z-10 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +11 z^8-22 z^6+6 z^4+5 z^2-3 (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 \
10          1-1
8         3 3
6        41 -3
4       73  4
2      75   -2
0     86    2
-2    68     2
-4   57      -2
-6  26       4
-8 15        -4
-10 2         2
-121          -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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