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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n81 at Knotilus!

Link Presentations

[edit Notes on L10n81's Link Presentations]

Planar diagram presentation X6172 X2,16,3,15 X3,10,4,11 X5,14,6,15 X11,20,12,13 X13,12,14,5 X19,1,20,4 X8,17,9,18 X16,7,17,8 X18,9,19,10
Gauss code {1, -2, -3, 7}, {-4, -1, 9, -8, 10, 3, -5, 6}, {-6, 4, 2, -9, 8, -10, -7, 5}
A Braid Representative
A Morse Link Presentation L10n81 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+2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)+2 t(1) t(3) t(2)-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  q^{-10} -2 q^{-9} +5 q^{-8} -6 q^{-7} +6 q^{-6} -6 q^{-5} +6 q^{-4} -2 q^{-3} +2 q^{-2} (db)
Signature -4 (db)
HOMFLY-PT polynomial a^8 z^4+3 a^8 z^2+a^8 z^{-2} +4 a^8-a^6 z^6-5 a^6 z^4-11 a^6 z^2-2 a^6 z^{-2} -11 a^6+2 a^4 z^4+7 a^4 z^2+a^4 z^{-2} +7 a^4 (db)
Kauffman polynomial a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-2 a^{11} z^3+3 a^{10} z^6-4 a^{10} z^4+2 a^{10} z^2+2 a^9 z^7-2 a^9 z^3+a^8 z^8+4 a^8 z^4-7 a^8 z^2-a^8 z^{-2} +5 a^8+3 a^7 z^7-5 a^7 z^5+9 a^7 z^3-11 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+12 a^6 z^4-20 a^6 z^2-2 a^6 z^{-2} +13 a^6+a^5 z^7-3 a^5 z^5+9 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-9 a^4 z^2-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       220
-7      4  4
-9     22  0
-11    44   0
-13   33    0
-15  23     -1
-17  3      3
-1912       -1
-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}^{2}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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

See/edit the Link_Splice_Base (expert).

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