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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 L10a135's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,5,15,6 X16,12,17,11 X10,13,5,14 X20,18,11,17 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15
Gauss code {1, -9, 2, -10}, {3, -1, 8, -7, 9, -5}, {4, -2, 5, -3, 10, -4, 6, -8, 7, -6}
A Braid Representative
A Morse Link Presentation L10a135 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^2 w^2-3 u v^2 w+u v^2-2 u v w^2+6 u v w-3 u v-2 u w+2 u-2 v^2 w^2+2 v^2 w+3 v w^2-6 v w+2 v-w^2+3 w-2}{\sqrt{u} v w} (db)
Jones polynomial q^6-4 q^5+7 q^4+ q^{-4} -10 q^3-3 q^{-3} +14 q^2+7 q^{-2} -13 q-10 q^{-1} +14 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2+z^2 a^{-2} +z^2 a^{-4} -5 z^2+2 a^2+4 a^{-2} - a^{-4} -5+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial z^6 a^{-6} -2 z^4 a^{-6} +z^2 a^{-6} +4 z^7 a^{-5} -11 z^5 a^{-5} +7 z^3 a^{-5} +z a^{-5} +5 z^8 a^{-4} -12 z^6 a^{-4} +a^4 z^4+6 z^4 a^{-4} -a^4 z^2+z^2 a^{-4} -2 a^{-4} +2 z^9 a^{-3} +5 z^7 a^{-3} +3 a^3 z^5-20 z^5 a^{-3} -2 a^3 z^3+8 z^3 a^{-3} +3 z a^{-3} +10 z^8 a^{-2} +6 a^2 z^6-18 z^6 a^{-2} -8 a^2 z^4+8 a^2 z^2+11 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-8 a^{-2} +2 z^9 a^{-1} +7 a z^7+8 z^7 a^{-1} -8 a z^5-20 z^5 a^{-1} +3 a z^3+6 z^3 a^{-1} +3 a z+5 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +5 z^8+z^6-17 z^4+20 z^2+2 z^{-2} -9 (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 \
13          11
11         3 -3
9        41 3
7       63  -3
5      84   4
3     67    1
1    87     1
-1   48      4
-3  36       -3
-5 15        4
-7 2         -2
-91          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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