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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 L10a155's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,14,17,13 X18,8,19,7 X10,18,11,17 X14,9,15,10 X8,15,9,16 X16,20,5,19 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {5, -4, 8, -3}, {9, -1, 4, -7, 6, -5, 10, -2, 3, -6, 7, -8}
A Braid Representative
A Morse Link Presentation L10a155 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(w-1) \left(u v w^2-3 u v w+2 u v+4 u w-2 u-2 v w^2+4 v w+2 w^2-3 w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial q^4-4 q^3+10 q^2-12 q+16-16 q^{-1} +15 q^{-2} -11 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6-3 z^2 a^4-2 a^4+3 z^4 a^2+5 z^2 a^2+a^2 z^{-2} +4 a^2-z^6-3 z^4-7 z^2-2 z^{-2} -6+z^4 a^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} (db)
Kauffman polynomial 2 a^3 z^9+2 a z^9+4 a^4 z^8+12 a^2 z^8+8 z^8+3 a^5 z^7+7 a^3 z^7+16 a z^7+12 z^7 a^{-1} +a^6 z^6-7 a^4 z^6-21 a^2 z^6+10 z^6 a^{-2} -3 z^6-8 a^5 z^5-27 a^3 z^5-38 a z^5-15 z^5 a^{-1} +4 z^5 a^{-3} -3 a^6 z^4+4 a^2 z^4-12 z^4 a^{-2} +z^4 a^{-4} -12 z^4+7 a^5 z^3+23 a^3 z^3+17 a z^3+z^3 a^{-1} +3 a^6 z^2+4 a^4 z^2+3 a^2 z^2+8 z^2 a^{-2} +10 z^2-2 a^5 z-6 a^3 z+2 a z+6 z a^{-1} -a^6-2 a^2-5 a^{-2} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} (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 \
9          11
7         41-3
5        6  6
3       64  -2
1      106   4
-1     88    0
-3    78     -1
-5   48      4
-7  37       -4
-9 15        4
-11 2         -2
-131          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z} {\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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