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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 L10a134's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X16,7,17,8 X20,9,11,10 X18,12,19,11 X8,15,9,16 X10,19,5,20 X14,18,15,17 X2536 X4,14,1,13
Gauss code {1, -9, 2, -10}, {9, -1, 3, -6, 4, -7}, {5, -2, 10, -8, 6, -3, 8, -5, 7, -4}
A Braid Representative
A Morse Link Presentation L10a134 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^2 w^2-2 u v^2 w+u v^2-3 u v w^2+5 u v w-2 u v+3 u w^2-3 u w+u-v^2 w^2+3 v^2 w-3 v^2+2 v w^2-5 v w+3 v-w^2+2 w-1}{\sqrt{u} v w} (db)
Jones polynomial  q^{-6} -2 q^{-5} +q^4+7 q^{-4} -4 q^3-9 q^{-3} +8 q^2+13 q^{-2} -11 q-14 q^{-1} +14 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 a^4 z^2-2 a^4 z^{-2} -4 a^4+3 a^2 z^4+z^4 a^{-2} +6 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +4 a^2+ a^{-2} -z^6-3 z^4-5 z^2-2 (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^7-4 a^5 z^5+4 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-5 a^4 z^6-a^4 z^4+z^4 a^{-4} +6 a^4 z^2+2 a^4 z^{-2} -6 a^4+a^3 z^9+9 a^3 z^7-28 a^3 z^5+4 z^5 a^{-3} +20 a^3 z^3-2 z^3 a^{-3} -2 a^3 z^{-1} +8 a^2 z^8-9 a^2 z^6+8 z^6 a^{-2} -9 a^2 z^4-9 z^4 a^{-2} +9 a^2 z^2+4 z^2 a^{-2} +a^2 z^{-2} -3 a^2- a^{-2} +a z^9+16 a z^7+9 z^7 a^{-1} -39 a z^5-11 z^5 a^{-1} +27 a z^3+5 z^3 a^{-1} -6 a z-2 z a^{-1} +5 z^8+5 z^6-22 z^4+13 z^2-1 (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 \
9          11
7         3 -3
5        51 4
3       63  -3
1      85   3
-1     88    0
-3    56     -1
-5   59      4
-7  24       -2
-9  5        5
-1112         -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} {\mathbb Z}
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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