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

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Link Presentations

[edit Notes on L11a479's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X18,7,19,8 X22,15,17,16 X20,10,21,9 X8,13,9,14 X14,17,15,18 X16,21,5,22 X12,20,13,19 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {7, -3, 9, -5, 8, -4}, {10, -1, 3, -6, 5, -2, 11, -9, 6, -7, 4, -8}
A Braid Representative
A Morse Link Presentation L11a479 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(3)-1)^3 (t(2)+t(3))}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial q^4-4 q^3+9 q^2-14 q+19-20 q^{-1} +21 q^{-2} -16 q^{-3} +13 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7} (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^6-a^6+2 z^4 a^4+3 z^2 a^4+a^4 z^{-2} +2 a^4-z^6 a^2-z^4 a^2-2 a^2 z^{-2} -a^2-z^6-2 z^4-3 z^2+ z^{-2} -1+z^4 a^{-2} +z^2 a^{-2} + a^{-2} (db)
Kauffman polynomial a^7 z^7-4 a^7 z^5+5 a^7 z^3-2 a^7 z+3 a^6 z^8-11 a^6 z^6+14 a^6 z^4-9 a^6 z^2+3 a^6+4 a^5 z^9-10 a^5 z^7+a^5 z^5+11 a^5 z^3-7 a^5 z+2 a^4 z^{10}+6 a^4 z^8-37 a^4 z^6+48 a^4 z^4+z^4 a^{-4} -27 a^4 z^2-a^4 z^{-2} +9 a^4+12 a^3 z^9-28 a^3 z^7+7 a^3 z^5+4 z^5 a^{-3} +15 a^3 z^3-z^3 a^{-3} -10 a^3 z+2 a^3 z^{-1} +2 a^2 z^{10}+16 a^2 z^8-54 a^2 z^6+9 z^6 a^{-2} +48 a^2 z^4-8 z^4 a^{-2} -20 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2- a^{-2} +8 a z^9-4 a z^7+13 z^7 a^{-1} -19 a z^5-17 z^5 a^{-1} +17 a z^3+7 z^3 a^{-1} -6 a z-z a^{-1} +2 a z^{-1} +13 z^8-19 z^6+5 z^4+z^2- z^{-2} +1 (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 \
9           11
7          3 -3
5         61 5
3        83  -5
1       116   5
-1      1110    -1
-3     109     1
-5    813      5
-7   58       -3
-9  28        6
-11 15         -4
-13 2          2
-151           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.

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