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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 L11a480's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X18,7,19,8 X14,21,15,22 X20,10,21,9 X8,13,9,14 X22,15,17,16 X16,17,5,18 X12,20,13,19 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {8, -3, 9, -5, 4, -7}, {10, -1, 3, -6, 5, -2, 11, -9, 6, -4, 7, -8}
A Braid Representative
A Morse Link Presentation L11a480 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)+1)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial q^3-4 q^2+9 q-13+19 q^{-1} -20 q^{-2} +21 q^{-3} -17 q^{-4} +13 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^6 z^4-3 a^6 z^2-2 a^6+2 a^4 z^6+8 a^4 z^4+11 a^4 z^2+a^4 z^{-2} +6 a^4-a^2 z^8-5 a^2 z^6-10 a^2 z^4-11 a^2 z^2-2 a^2 z^{-2} -7 a^2+z^6+3 z^4+3 z^2+ z^{-2} +3 (db)
Kauffman polynomial 2 a^4 z^{10}+2 a^2 z^{10}+5 a^5 z^9+12 a^3 z^9+7 a z^9+6 a^6 z^8+10 a^4 z^8+12 a^2 z^8+8 z^8+5 a^7 z^7-2 a^5 z^7-26 a^3 z^7-15 a z^7+4 z^7 a^{-1} +3 a^8 z^6-6 a^6 z^6-32 a^4 z^6-47 a^2 z^6+z^6 a^{-2} -23 z^6+a^9 z^5-6 a^7 z^5-5 a^5 z^5+18 a^3 z^5+7 a z^5-9 z^5 a^{-1} -5 a^8 z^4+a^6 z^4+39 a^4 z^4+56 a^2 z^4-2 z^4 a^{-2} +21 z^4-2 a^9 z^3+a^7 z^3+6 a^5 z^3+a^3 z^3+a z^3+3 z^3 a^{-1} +3 a^8 z^2-23 a^4 z^2-30 a^2 z^2-10 z^2+a^9 z+a^7 z-3 a^5 z-6 a^3 z-3 a z-a^8+7 a^4+9 a^2+4+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          3 -3
3         61 5
1        73  -4
-1       126   6
-3      109    -1
-5     1110     1
-7    812      4
-9   59       -4
-11  28        6
-13 15         -4
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 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}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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.

Modifying This Page

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