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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 L11a382's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,7,17,8 X18,10,19,9 X20,6,21,5 X6,22,7,21 X22,18,11,17 X10,20,1,19 X8,11,9,12 X4,15,5,16
Gauss code {1, -2, 3, -11, 6, -7, 4, -10, 5, -9}, {10, -1, 2, -3, 11, -4, 8, -5, 9, -6, 7, -8}
A Braid Representative
A Morse Link Presentation L11a382 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(t(1)^2 t(2)^2-t(1) t(2)^2+2 t(1) t(2)-t(2)-t(1)+1\right) \left(t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+2 t(2) t(1)-t(1)+1\right)}{t(1)^2 t(2)^2} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+13 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a z^9+a^3 z^7-7 a z^7+z^7 a^{-1} +5 a^3 z^5-18 a z^5+5 z^5 a^{-1} +8 a^3 z^3-20 a z^3+8 z^3 a^{-1} +4 a^3 z-9 a z+4 z a^{-1} +a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial a^7 z^5-2 a^7 z^3+a^7 z+3 a^6 z^6-6 a^6 z^4+3 a^6 z^2+4 a^5 z^7-5 a^5 z^5-a^5 z^3+a^5 z+4 a^4 z^8-3 a^4 z^6+z^6 a^{-4} -3 a^4 z^4-3 z^4 a^{-4} +a^4 z^2+z^2 a^{-4} +4 a^3 z^9-8 a^3 z^7+3 z^7 a^{-3} +14 a^3 z^5-9 z^5 a^{-3} -14 a^3 z^3+5 z^3 a^{-3} +5 a^3 z-z a^{-3} -a^3 z^{-1} +2 a^2 z^{10}+5 z^8 a^{-2} -6 a^2 z^6-16 z^6 a^{-2} +12 a^2 z^4+14 z^4 a^{-2} -6 a^2 z^2-5 z^2 a^{-2} +a^2+9 a z^9+5 z^9 a^{-1} -31 a z^7-16 z^7 a^{-1} +47 a z^5+18 z^5 a^{-1} -30 a z^3-10 z^3 a^{-1} +9 a z+3 z a^{-1} -a z^{-1} +2 z^{10}+z^8-17 z^6+26 z^4-10 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 \
10           1-1
8          2 2
6         41 -3
4        62  4
2       74   -3
0      96    3
-2     78     1
-4    78      -1
-6   48       4
-8  26        -4
-10 14         3
-12 2          -2
-141           1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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