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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 L11a446's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X18,9,19,10 X16,8,17,7 X22,17,13,18 X20,11,21,12 X8,14,9,13 X10,19,11,20 X12,21,5,22 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 4, -7, 3, -8, 6, -9}, {7, -2, 11, -4, 5, -3, 8, -6, 9, -5}
A Braid Representative
A Morse Link Presentation L11a446 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^3 w^3-u v^3 w^2-2 u v^2 w^3+4 u v^2 w^2-2 u v^2 w+u v w^3-3 u v w^2+4 u v w-u v+u w^2-2 u w+u-v^3 w^3+2 v^3 w^2-v^3 w+v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+2 v w^2-4 v w+2 v+w-1}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial q^3-3 q^2+6 q-8+13 q^{-1} -14 q^{-2} +15 q^{-3} -12 q^{-4} +10 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^2 z^8+2 a^4 z^6-6 a^2 z^6+z^6-a^6 z^4+9 a^4 z^4-14 a^2 z^4+4 z^4-3 a^6 z^2+13 a^4 z^2-16 a^2 z^2+5 z^2-2 a^6+7 a^4-9 a^2+4+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-6 a^8 z^4+3 a^8 z^2-a^8+4 a^7 z^7-5 a^7 z^5-2 a^7 z^3+a^7 z+4 a^6 z^8-4 a^6 z^6-2 a^6 z^4+a^6 z^2+3 a^5 z^9-2 a^5 z^7-3 a^5 z^5+6 a^5 z^3-3 a^5 z+a^4 z^{10}+7 a^4 z^8-29 a^4 z^6+47 a^4 z^4-31 a^4 z^2-a^4 z^{-2} +9 a^4+7 a^3 z^9-20 a^3 z^7+19 a^3 z^5+4 a^3 z^3-8 a^3 z+2 a^3 z^{-1} +a^2 z^{10}+8 a^2 z^8-41 a^2 z^6+z^6 a^{-2} +67 a^2 z^4-3 z^4 a^{-2} -44 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+4 a z^9-11 a z^7+3 z^7 a^{-1} +7 a z^5-9 z^5 a^{-1} +2 a z^3+4 z^3 a^{-1} -5 a z+2 a z^{-1} +5 z^8-18 z^6+21 z^4-14 z^2- z^{-2} +6 (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          2 -2
3         41 3
1        42  -2
-1       94   5
-3      87    -1
-5     76     1
-7    58      3
-9   57       -2
-11  26        4
-13 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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