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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 L11n306's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X15,22,16,11 X13,20,14,21 X21,14,22,15 X17,8,18,9 X7,16,8,17 X9,18,10,19 X19,10,20,5 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {10, -1, -7, 6, -8, 9}, {-11, 2, -4, 5, -3, 7, -6, 8, -9, 4, -5, 3}
A Braid Representative
A Morse Link Presentation L11n306 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^2 w^4-u v^2 w^3+u v^2 w^2-u v^2 w+u v w-u v+u-v^2 w^4+v w^4-v w^3+w^3-w^2+w-1}{\sqrt{u} v w^2} (db)
Jones polynomial - q^{-10} +2 q^{-9} -3 q^{-8} +4 q^{-7} -4 q^{-6} +5 q^{-5} -3 q^{-4} +4 q^{-3} - q^{-2} + q^{-1} (db)
Signature -6 (db)
HOMFLY-PT polynomial a^{10} \left(-z^2\right)-a^{10} z^{-2} -2 a^{10}+a^8 z^6+6 a^8 z^4+12 a^8 z^2+4 a^8 z^{-2} +11 a^8-a^6 z^8-7 a^6 z^6-18 a^6 z^4-24 a^6 z^2-5 a^6 z^{-2} -18 a^6+a^4 z^6+6 a^4 z^4+12 a^4 z^2+2 a^4 z^{-2} +9 a^4 (db)
Kauffman polynomial z a^{13}+2 z^2 a^{12}-a^{12}+z^5 a^{11}-z^3 a^{11}-2 z a^{11}+a^{11} z^{-1} +3 z^6 a^{10}-10 z^4 a^{10}+6 z^2 a^{10}-a^{10} z^{-2} +5 z^7 a^9-23 z^5 a^9+31 z^3 a^9-19 z a^9+5 a^9 z^{-1} +4 z^8 a^8-19 z^6 a^8+27 z^4 a^8-21 z^2 a^8-4 a^8 z^{-2} +13 a^8+z^9 a^7+z^7 a^7-25 z^5 a^7+50 z^3 a^7-35 z a^7+9 a^7 z^{-1} +5 z^8 a^6-29 z^6 a^6+55 z^4 a^6-46 z^2 a^6-5 a^6 z^{-2} +22 a^6+z^9 a^5-4 z^7 a^5-z^5 a^5+18 z^3 a^5-19 z a^5+5 a^5 z^{-1} +z^8 a^4-7 z^6 a^4+18 z^4 a^4-21 z^2 a^4-2 a^4 z^{-2} +11 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       41 3
-7      12  1
-9     42   2
-11   122    1
-13   33     0
-15 122      1
-17 23       -1
-19 1        1
-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}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3} {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
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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