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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 L11n452's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,20,12,21 X13,19,14,22 X21,18,22,9 X17,13,18,12 X8,16,5,15 X14,8,15,7 X19,17,20,16 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 8, -7}, {-9, 3, -5, 4}, {11, -2, -3, 6, -4, -8, 7, 9, -6, 5}
A Braid Representative
A Morse Link Presentation L11n452 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) (t(4)-1)^2 (t(2) t(4)-t(1))}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} (db)
Jones polynomial q^{9/2}+\frac{2}{q^{9/2}}-q^{7/2}-\frac{6}{q^{7/2}}-q^{5/2}+\frac{5}{q^{5/2}}+2 q^{3/2}-\frac{7}{q^{3/2}}-\frac{1}{q^{11/2}}-6 \sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z^{-3} +a^5 z+2 a^5 z^{-1} -3 a^3 z^3-3 a^3 z^{-3} +z^3 a^{-3} -8 a^3 z-8 a^3 z^{-1} +3 z a^{-3} + a^{-3} z^{-1} +2 a z^5-z^5 a^{-1} +9 a z^3+3 a z^{-3} -7 z^3 a^{-1} - a^{-1} z^{-3} +15 a z+11 a z^{-1} -11 z a^{-1} -6 a^{-1} z^{-1} (db)
Kauffman polynomial a^5 z^7-5 a^5 z^5+10 a^5 z^3-a^5 z^{-3} -10 a^5 z+5 a^5 z^{-1} +2 a^4 z^8-7 a^4 z^6+z^6 a^{-4} +3 a^4 z^4-5 z^4 a^{-4} +7 a^4 z^2+4 z^2 a^{-4} +3 a^4 z^{-2} -9 a^4- a^{-4} +a^3 z^9+3 a^3 z^7+z^7 a^{-3} -29 a^3 z^5-7 z^5 a^{-3} +45 a^3 z^3+11 z^3 a^{-3} -3 a^3 z^{-3} -34 a^3 z-9 z a^{-3} +14 a^3 z^{-1} +2 a^{-3} z^{-1} +6 a^2 z^8-23 a^2 z^6+12 a^2 z^4-7 z^4 a^{-2} +19 a^2 z^2+15 z^2 a^{-2} +6 a^2 z^{-2} -21 a^2-6 a^{-2} +a z^9+5 a z^7+4 z^7 a^{-1} -44 a z^5-27 z^5 a^{-1} +75 a z^3+51 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -50 a z-35 z a^{-1} +18 a z^{-1} +11 a^{-1} z^{-1} +4 z^8-17 z^6+7 z^4+23 z^2+3 z^{-2} -18 (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            0
6        211 2
4       21   -1
2      521   4
0     581    2
-2    234     3
-4   35       2
-6  32        1
-8 15         4
-10 1          -1
-121           1
Integral Khovanov Homology

(db, data source)

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

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