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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 L11n202's Link Presentations]

Planar diagram presentation X10,1,11,2 X16,8,17,7 X11,18,12,19 X19,3,20,2 X3,12,4,13 X13,21,14,20 X5,15,6,14 X6,9,7,10 X15,22,16,9 X8,18,1,17 X21,4,22,5
Gauss code {1, 4, -5, 11, -7, -8, 2, -10}, {8, -1, -3, 5, -6, 7, -9, -2, 10, 3, -4, 6, -11, 9}
A Braid Representative
A Morse Link Presentation L11n202 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(u^2 v^2+u v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{7/2}-2 q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^3 z^5+4 a^3 z^3+3 a^3 z-a z^7-6 a z^5+z^5 a^{-1} -11 a z^3+4 z^3 a^{-1} -6 a z+3 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^5 z^7-5 a^5 z^5+6 a^5 z^3-a^5 z+2 a^4 z^8-11 a^4 z^6+17 a^4 z^4-7 a^4 z^2+z^2 a^{-4} +a^3 z^9-4 a^3 z^7+3 a^3 z^5-a^3 z^3+2 z^3 a^{-3} +2 a^3 z-z a^{-3} +3 a^2 z^8-15 a^2 z^6+22 a^2 z^4+2 z^4 a^{-2} -11 a^2 z^2-z^2 a^{-2} +a z^9-5 a z^7+11 a z^5+3 z^5 a^{-1} -15 a z^3-6 z^3 a^{-1} +6 a z+2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-4 z^6+7 z^4-6 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 \
8         1-1
6        1 1
4       11 0
2      31  2
0     23   1
-2    21    1
-4   12     1
-6  12      -1
-8 11       0
-10 1        -1
-121         1
Integral Khovanov Homology

(db, data source)

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