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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 L11n425's Link Presentations]

Planar diagram presentation X8192 X16,8,17,7 X5,14,6,15 X3,10,4,11 X13,4,14,5 X17,2,18,3 X9,19,10,18 X21,7,22,12 X11,13,12,22 X20,16,21,15 X6,19,1,20
Gauss code {1, 6, -4, 5, -3, -11}, {2, -1, -7, 4, -9, 8}, {-5, 3, 10, -2, -6, 7, 11, -10, -8, 9}
A Braid Representative
A Morse Link Presentation L11n425 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(2)^2 t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3+t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2+t(1)^2 t(3)-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)+t(1)+t(1)^2 t(2)-t(1) t(2)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial 2 q^2-3 q+5-5 q^{-1} +6 q^{-2} -4 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^2 z^6+a^4 z^4-5 a^2 z^4+z^4+3 a^4 z^2-9 a^2 z^2+2 z^2+3 a^4-6 a^2+ a^{-2} +2+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+3 a^4 z^8+4 a^2 z^8+z^8+2 a^5 z^7-a^3 z^7-3 a z^7+a^6 z^6-14 a^4 z^6-19 a^2 z^6-4 z^6-7 a^5 z^5-8 a^3 z^5+z^5 a^{-1} -4 a^6 z^4+22 a^4 z^4+33 a^2 z^4+7 z^4+4 a^5 z^3+14 a^3 z^3+10 a z^3+3 a^6 z^2-19 a^4 z^2-27 a^2 z^2+2 z^2 a^{-2} -3 z^2-9 a^3 z-9 a z+7 a^4+11 a^2-2 a^{-2} +3+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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 \
5        22
3       21-1
1      31 2
-1     44  0
-3    21   1
-5   24    2
-7  22     0
-9  2      2
-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}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}

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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