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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 L11n323's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X11,21,12,20 X19,22,20,13 X13,12,14,5 X4,17,1,18 X21,11,22,10
Gauss code {1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11, -7, 9}, {-9, 2, -4, -5, 10, 6, -8, 7, -11, 8}
A Braid Representative
A Morse Link Presentation L11n323 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+t(2)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial - q^{-10} + q^{-9} +2 q^{-7} - q^{-6} +2 q^{-5} - q^{-4} +2 q^{-3} - q^{-2} + q^{-1} (db)
Signature -6 (db)
HOMFLY-PT polynomial a^{10} \left(-z^2\right)-2 a^{10}+a^8 z^6+6 a^8 z^4+10 a^8 z^2+a^8 z^{-2} +6 a^8-a^6 z^8-7 a^6 z^6-16 a^6 z^4-16 a^6 z^2-2 a^6 z^{-2} -9 a^6+a^4 z^6+6 a^4 z^4+10 a^4 z^2+a^4 z^{-2} +5 a^4 (db)
Kauffman polynomial z a^{13}+z^2 a^{12}-a^{12}+z a^{11}+z^4 a^{10}-3 z^2 a^{10}+z^7 a^9-5 z^5 a^9+6 z^3 a^9-3 z a^9+2 z^8 a^8-13 z^6 a^8+27 z^4 a^8-24 z^2 a^8-a^8 z^{-2} +9 a^8+z^9 a^7-5 z^7 a^7+4 z^5 a^7+6 z^3 a^7-8 z a^7+2 a^7 z^{-1} +3 z^8 a^6-20 z^6 a^6+42 z^4 a^6-35 z^2 a^6-2 a^6 z^{-2} +13 a^6+z^9 a^5-6 z^7 a^5+9 z^5 a^5-5 z a^5+2 a^5 z^{-1} +z^8 a^4-7 z^6 a^4+16 z^4 a^4-15 z^2 a^4-a^4 z^{-2} +6 a^4 (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 \
-1         11
-3          0
-5       21 1
-7     111  1
-9     21   1
-11   221    1
-13  142     1
-15 112      2
-17 21       1
-1911        0
-211         -1
Integral Khovanov Homology

(db, data source)

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