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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 L11n292's Link Presentations]

Planar diagram presentation X6172 X11,18,12,19 X3849 X2,16,3,15 X16,7,17,8 X9,11,10,22 X17,1,18,4 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 L11n292 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(3)^4+t(1) t(2) t(3)^4-t(2)^2 t(3)^3+2 t(1) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+2 t(2)^2 t(3)^2-2 t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2-2 t(2)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)+t(2)^2-t(2)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial q^5-3 q^4- q^{-4} +5 q^3+3 q^{-3} -6 q^2-4 q^{-2} +8 q+7 q^{-1} -6 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^2 a^{-4} + a^{-4} -a^2 z^4-2 z^4 a^{-2} -2 a^2 z^2-5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^{-2} +z^6+4 z^4+5 z^2-2 z^{-2} +1 (db)
Kauffman polynomial 2 a z^9+2 z^9 a^{-1} +3 a^2 z^8+5 z^8 a^{-2} +8 z^8+a^3 z^7-5 a z^7-2 z^7 a^{-1} +4 z^7 a^{-3} -14 a^2 z^6-21 z^6 a^{-2} +z^6 a^{-4} -36 z^6-4 a^3 z^5-6 a z^5-16 z^5 a^{-1} -14 z^5 a^{-3} +19 a^2 z^4+30 z^4 a^{-2} +z^4 a^{-4} +48 z^4+4 a^3 z^3+14 a z^3+24 z^3 a^{-1} +17 z^3 a^{-3} +3 z^3 a^{-5} -9 a^2 z^2-20 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -25 z^2-a^3 z-3 a z-7 z a^{-1} -7 z a^{-3} -2 z a^{-5} +4 a^{-2} +2 a^{-4} +3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +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 \
11         11
9        2 -2
7       31 2
5      43  -1
3     431  2
1    46    2
-1   331    1
-3  25      3
-5 12       -1
-7 2        2
-91         -1
Integral Khovanov Homology

(db, data source)

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