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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 L10a128's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,14,11,13 X18,16,19,15 X14,8,15,7 X10,20,5,19 X8,17,9,18 X16,9,17,10 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {9, -1, 5, -7, 8, -6}, {10, -2, 3, -5, 4, -8, 7, -4, 6, -3}
A Braid Representative
A Morse Link Presentation L10a128 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+t(1) t(2)^2-2 t(1) t(3) t(2)^2+3 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+3 t(3)^2 t(2)-3 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+t(1) t(3)^2-t(3)^2+2 t(1)-3 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial  q^{-6} -2 q^{-5} +q^4+6 q^{-4} -4 q^3-8 q^{-3} +7 q^2+12 q^{-2} -10 q-12 q^{-1} +13 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 a^4 z^2-2 a^4 z^{-2} -5 a^4+3 a^2 z^4+z^4 a^{-2} +7 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +5 a^2-z^6-3 z^4-4 z^2-1 (db)
Kauffman polynomial a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-5 a^5 z^5+2 a^5 z^3+3 a^5 z-2 a^5 z^{-1} +2 a^4 z^8-13 a^4 z^4+z^4 a^{-4} +18 a^4 z^2+2 a^4 z^{-2} -9 a^4+a^3 z^9+5 a^3 z^7-14 a^3 z^5+4 z^5 a^{-3} +4 a^3 z^3-3 z^3 a^{-3} +5 a^3 z-2 a^3 z^{-1} +6 a^2 z^8-4 a^2 z^6+7 z^6 a^{-2} -16 a^2 z^4-8 z^4 a^{-2} +19 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} -8 a^2+a z^9+10 a z^7+7 z^7 a^{-1} -19 a z^5-6 z^5 a^{-1} +4 a z^3-z^3 a^{-1} +3 a z+z a^{-1} +4 z^8+4 z^6-16 z^4+10 z^2-2 (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         3 -3
5        41 3
3       63  -3
1      74   3
-1     67    1
-3    66     0
-5   48      4
-7  24       -2
-9 15        4
-11 1         -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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