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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 L11a533's Link Presentations]

Planar diagram presentation X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X22,16,13,15 X20,18,21,17 X16,22,17,21 X12,20,9,19
Gauss code {1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 8, -10, 9, -3, 11, -9, 10, -8}
A Braid Representative
A Morse Link Presentation L11a533 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 t(2) t(1)-2 t(2) t(3) t(1)+2 t(3) t(1)-2 t(2) t(4) t(1)+2 t(2) t(3) t(4) t(1)-3 t(3) t(4) t(1)+3 t(4) t(1)-3 t(1)-3 t(2)+3 t(2) t(3)-2 t(3)+2 t(2) t(4)-3 t(2) t(3) t(4)+3 t(3) t(4)-2 t(4)+2}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+7 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} +a^7 z^{-3} -3 z a^5-5 a^5 z^{-1} -3 a^5 z^{-3} +3 z^3 a^3+6 z a^3+7 a^3 z^{-1} +3 a^3 z^{-3} -z^5 a-z^3 a-3 z a-3 a z^{-1} -a z^{-3} -z^5 a^{-1} -z^3 a^{-1} +z^3 a^{-3} (db)
Kauffman polynomial a^7 z^5-4 a^7 z^3+a^7 z^{-3} +6 a^7 z-4 a^7 z^{-1} +a^6 z^6-6 a^6 z^2-3 a^6 z^{-2} +8 a^6+a^5 z^7+3 a^5 z^5-12 a^5 z^3+3 a^5 z^{-3} +14 a^5 z-9 a^5 z^{-1} +a^4 z^8+2 a^4 z^6+z^6 a^{-4} -2 z^4 a^{-4} -12 a^4 z^2-6 a^4 z^{-2} +15 a^4+a^3 z^9+a^3 z^7+4 z^7 a^{-3} +3 a^3 z^5-11 z^5 a^{-3} -12 a^3 z^3+4 z^3 a^{-3} +3 a^3 z^{-3} +14 a^3 z-9 a^3 z^{-1} +a^2 z^{10}-a^2 z^8+6 z^8 a^{-2} +5 a^2 z^6-19 z^6 a^{-2} -3 a^2 z^4+13 z^4 a^{-2} -6 a^2 z^2-3 a^2 z^{-2} +8 a^2+5 a z^9+4 z^9 a^{-1} -14 a z^7-10 z^7 a^{-1} +17 a z^5+5 z^5 a^{-1} -8 a z^3+a z^{-3} +6 a z-4 a z^{-1} +z^{10}+4 z^8-16 z^6+12 z^4 (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          3 3
6         41 -3
4        53  2
2       64   -2
0      85    3
-2     710     3
-4    44      0
-6   17       6
-8  44        0
-10 15         4
-12            0
-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 {\mathbb Z}
r=-4 {\mathbb Z}^{5} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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