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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 L11n136's Link Presentations]

Planar diagram presentation X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X19,4,20,5 X14,22,15,21
Gauss code {1, -4, -3, 10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, 4, 2, -10, -8, 11, -9}
A Braid Representative
A Morse Link Presentation L11n136 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^6-t(1) t(2)^5+t(2)^5+t(1) t(2)^4-t(1) t(2)^3+t(1) t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(1)}{t(1) t(2)^3} (db)
Jones polynomial \frac{1}{q^{9/2}}+q^{7/2}-\frac{2}{q^{7/2}}-q^{5/2}+\frac{2}{q^{5/2}}+2 q^{3/2}-\frac{3}{q^{3/2}}-\frac{1}{q^{11/2}}-3 \sqrt{q}+\frac{2}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^3 z^5+5 a^3 z^3+7 a^3 z+3 a^3 z^{-1} -a z^7-7 a z^5+z^5 a^{-1} -17 a z^3+5 z^3 a^{-1} -17 a z+7 z a^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-a^4 z^8-2 a^2 z^8-z^8-a^5 z^7+6 a^3 z^7+7 a z^7+5 a^4 z^6+11 a^2 z^6+6 z^6+6 a^5 z^5-13 a^3 z^5-21 a z^5-2 z^5 a^{-1} -6 a^4 z^4-20 a^2 z^4-z^4 a^{-2} -15 z^4-10 a^5 z^3+16 a^3 z^3+33 a z^3+6 z^3 a^{-1} -z^3 a^{-3} +a^4 z^2+17 a^2 z^2-z^2 a^{-4} +17 z^2+4 a^5 z-11 a^3 z-22 a z-7 z a^{-1} -5 a^2+ a^{-4} -5+3 a^3 z^{-1} +5 a z^{-1} +2 a^{-1} z^{-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          0
4       21 -1
2      1   1
0     23   1
-2    1     1
-4   12     1
-6  11      0
-8  1       1
-1011        0
-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} {\mathbb Z}
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3} {\mathbb Z}
r=1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=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

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

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