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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 L11n291's Link Presentations]

Planar diagram presentation X6172 X11,18,12,19 X3849 X15,2,16,3 X16,7,17,8 X9,11,10,22 X4,17,1,18 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20
Gauss code {1, 4, -3, -7}, {-9, -1, 5, 3, -6, 8}, {-2, 9, -11, 10, -4, -5, 7, 2, -8, 11, -10, 6}
A Braid Representative
A Morse Link Presentation L11n291 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(2)^2 t(3)^4+t(1) t(2) t(3)^4+2 t(1) t(2)^2 t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3-t(3)^3-2 t(1) t(2)^2 t(3)^2+3 t(1) t(2) t(3)^2-3 t(2) t(3)^2+2 t(3)^2+t(1) t(2)^2 t(3)-t(1) t(2) t(3)+2 t(2) t(3)-2 t(3)-t(2)+1}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial q^3-3 q^2+5 q-7+9 q^{-1} -8 q^{-2} +9 q^{-3} -5 q^{-4} +4 q^{-5} - q^{-6} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^2 z^8+a^4 z^6-6 a^2 z^6+z^6+5 a^4 z^4-12 a^2 z^4+4 z^4-a^6 z^2+7 a^4 z^2-9 a^2 z^2+4 z^2-a^6+a^4-a^2+1+a^6 z^{-2} -2 a^4 z^{-2} +a^2 z^{-2} (db)
Kauffman polynomial 2 a^3 z^9+2 a z^9+4 a^4 z^8+8 a^2 z^8+4 z^8+2 a^5 z^7-3 a^3 z^7-2 a z^7+3 z^7 a^{-1} -17 a^4 z^6-32 a^2 z^6+z^6 a^{-2} -14 z^6-5 a^5 z^5-7 a^3 z^5-12 a z^5-10 z^5 a^{-1} +4 a^6 z^4+32 a^4 z^4+44 a^2 z^4-3 z^4 a^{-2} +13 z^4+a^7 z^3+9 a^5 z^3+19 a^3 z^3+18 a z^3+7 z^3 a^{-1} -5 a^6 z^2-21 a^4 z^2-24 a^2 z^2+z^2 a^{-2} -7 z^2-a^7 z-3 a^5 z-7 a^3 z-7 a z-2 z a^{-1} +3 a^4+4 a^2+2-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^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 \
7         11
5        2 -2
3       31 2
1      42  -2
-1     53   2
-3    45    1
-5   54     1
-7  26      4
-9 23       -1
-11 3        3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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