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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 L10a157's Link Presentations]

Planar diagram presentation X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,13,10 X12,18,7,17 X4,13,5,14 X14,5,15,6 X6,15,1,16
Gauss code {1, -4, 3, -8, 9, -10}, {5, -1, 6, -3, 2, -7}, {8, -9, 10, -5, 7, -2, 4, -6}
A Braid Representative
A Morse Link Presentation L10a157 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(u^2 v^2 w+u^2 v w^2-u^2 v w-u^2 w^2-u v^2 w+u v^2-u v w^2+u v w-u v+u w^2-u w-v^2-v w+v+w\right)}{u v w^{3/2}} (db)
Jones polynomial - q^{-6} +3 q^{-5} -q^4-5 q^{-4} +3 q^3+8 q^{-3} -4 q^2-9 q^{-2} +8 q+10 q^{-1} -8 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^2 z^6+z^6-a^4 z^4+3 a^2 z^4-z^4 a^{-2} +3 z^4-2 a^4 z^2+2 a^2 z^2-2 z^2 a^{-2} +z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial a^7 z^3+3 a^6 z^4-a^6 z^2+5 a^5 z^5-3 a^5 z^3+7 a^4 z^6-10 a^4 z^4+4 a^4 z^2+7 a^3 z^7+z^7 a^{-3} -13 a^3 z^5-4 z^5 a^{-3} +5 a^3 z^3+4 z^3 a^{-3} +5 a^2 z^8+3 z^8 a^{-2} -9 a^2 z^6-14 z^6 a^{-2} -a^2 z^4+20 z^4 a^{-2} +3 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +2 a z^9+2 z^9 a^{-1} +a z^7-5 z^7 a^{-1} -16 a z^5-2 z^5 a^{-1} +11 a z^3+6 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +8 z^8-30 z^6+32 z^4-10 z^2+2 z^{-2} -3 (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          1-1
7         2 2
5        21 -1
3       62  4
1      33   0
-1     75    2
-3    45     1
-5   45      -1
-7  25       3
-9 13        -2
-11 2         2
-131          -1
Integral Khovanov Homology

(db, data source)

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