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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 L11n134's Link Presentations]

Planar diagram presentation X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 X20,14,21,13 X22,16,7,15 X4,20,5,19 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 L11n134 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-u^2 v^3+u^2 v^2-u^2 v-u v^4+2 u v^3-u v^2+2 u v-u-v^3+v^2-v+1}{u v^2} (db)
Jones polynomial q^{7/2}-2 q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -a z^7+a^3 z^5-6 a z^5+z^5 a^{-1} +4 a^3 z^3-12 a z^3+4 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^3 z^9-a z^9-2 a^4 z^8-3 a^2 z^8-z^8-a^5 z^7+3 a^3 z^7+4 a z^7+10 a^4 z^6+12 a^2 z^6+2 z^6+5 a^5 z^5-10 a z^5-5 z^5 a^{-1} -14 a^4 z^4-13 a^2 z^4-3 z^4 a^{-2} -2 z^4-7 a^5 z^3-a^3 z^3+19 a z^3+11 z^3 a^{-1} -2 z^3 a^{-3} +5 a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} -z^2 a^{-4} +4 z^2+2 a^5 z-3 a^3 z-11 a z-6 z a^{-1} -a^2+a^3 z^{-1} +a 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        1 1
4       21 -1
2      31  2
0     23   1
-2    32    1
-4   23     1
-6  12      -1
-8 12       1
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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