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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 L11a432's Link Presentations]

Planar diagram presentation X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,14,19,13 X20,9,21,10 X12,19,5,20 X22,11,13,12 X10,21,11,22 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 L11a432 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(3)-1) (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial - q^{-8} +5 q^{-7} -10 q^{-6} +16 q^{-5} -20 q^{-4} +q^3+24 q^{-3} -4 q^2-22 q^{-2} +8 q+20 q^{-1} -13 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+6 a^4 z^4-9 a^2 z^4+3 z^4-a^6 z^2+3 a^4 z^2-4 a^2 z^2+2 z^2+a^6-4 a^4+3 a^2+a^6 z^{-2} -2 a^4 z^{-2} +a^2 z^{-2} (db)
Kauffman polynomial a^9 z^5+5 a^8 z^6-5 a^8 z^4+a^8+10 a^7 z^7-14 a^7 z^5+3 a^7 z^3+11 a^6 z^8-14 a^6 z^6+3 a^6 z^4-2 a^6 z^2+a^6 z^{-2} -a^6+7 a^5 z^9+a^5 z^7-16 a^5 z^5+5 a^5 z^3+4 a^5 z-2 a^5 z^{-1} +2 a^4 z^{10}+17 a^4 z^8-44 a^4 z^6+36 a^4 z^4-9 a^4 z^2+2 a^4 z^{-2} -4 a^4+13 a^3 z^9-22 a^3 z^7+7 a^3 z^5+3 a^3 z^3+4 a^3 z-2 a^3 z^{-1} +2 a^2 z^{10}+13 a^2 z^8-45 a^2 z^6+z^6 a^{-2} +47 a^2 z^4-2 z^4 a^{-2} -13 a^2 z^2+a^2 z^{-2} -3 a^2+6 a z^9-9 a z^7+4 z^7 a^{-1} -2 a z^5-10 z^5 a^{-1} +6 a z^3+5 z^3 a^{-1} +7 z^8-19 z^6+17 z^4-6 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         51 4
1        83  -5
-1       125   7
-3      1210    -2
-5     1210     2
-7    812      4
-9   812       -4
-11  410        6
-13 16         -5
-15 4          4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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