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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 L10n78's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X17,1,18,4 X19,10,20,11
Gauss code {1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10, -7, 8}, {-8, 2, 4, -5, -9, 6, -10, 7}
A Braid Representative
A Morse Link Presentation L10n78 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2+t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)-t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial 2 q^{-2} - q^{-3} +3 q^{-4} -3 q^{-5} +4 q^{-6} -3 q^{-7} +2 q^{-8} - q^{-9} + q^{-10} (db)
Signature -4 (db)
HOMFLY-PT polynomial a^8 z^4+4 a^8 z^2+a^8 z^{-2} +4 a^8-a^6 z^6-6 a^6 z^4-13 a^6 z^2-2 a^6 z^{-2} -11 a^6+2 a^4 z^4+8 a^4 z^2+a^4 z^{-2} +7 a^4 (db)
Kauffman polynomial z^4 a^{12}-3 z^2 a^{12}+a^{12}+z^5 a^{11}-2 z^3 a^{11}+z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+z^7 a^9-3 z^5 a^9+4 z^3 a^9+z^8 a^8-5 z^6 a^8+11 z^4 a^8-8 z^2 a^8-a^8 z^{-2} +5 a^8+2 z^7 a^7-9 z^5 a^7+17 z^3 a^7-11 z a^7+2 a^7 z^{-1} +z^8 a^6-6 z^6 a^6+17 z^4 a^6-23 z^2 a^6-2 a^6 z^{-2} +13 a^6+z^7 a^5-5 z^5 a^5+11 z^3 a^5-11 z a^5+2 a^5 z^{-1} +3 z^4 a^4-11 z^2 a^4-a^4 z^{-2} +8 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 \
-3        22
-5       121
-7      2  2
-9     11  0
-11    32   1
-13   12    1
-15  12     -1
-17  1      1
-1911       0
-211        1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-8 {\mathbb Z} {\mathbb Z}
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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