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

Visit L10a66 at Knotilus!

Link Presentations

[edit Notes on L10a66's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X6718 X16,11,17,12 X14,6,15,5 X4,16,5,15 X20,17,7,18 X18,14,19,13 X12,20,13,19
Gauss code {1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8}
A Braid Representative
A Morse Link Presentation L10a66 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1)^2 t(2)^4+t(1) t(2)^4+3 t(1)^2 t(2)^3-3 t(1) t(2)^3+t(2)^3-3 t(1)^2 t(2)^2+3 t(1) t(2)^2-3 t(2)^2+t(1)^2 t(2)-3 t(1) t(2)+3 t(2)+t(1)-1}{t(1) t(2)^2} (db)
Jones polynomial -q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^5+3 a^5 z^3+2 a^5 z+a^5 z^{-1} -a^3 z^7-5 a^3 z^5-9 a^3 z^3-8 a^3 z-2 a^3 z^{-1} +2 a z^5+8 a z^3-z^3 a^{-1} +8 a z-3 z a^{-1} +2 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -z^3 a^9-3 z^4 a^8+z^2 a^8-5 z^5 a^7+4 z^3 a^7-z a^7-6 z^6 a^6+7 z^4 a^6-z^2 a^6-6 z^7 a^5+11 z^5 a^5-5 z^3 a^5+2 z a^5-a^5 z^{-1} -4 z^8 a^4+7 z^6 a^4-z^9 a^3-6 z^7 a^3+29 z^5 a^3-31 z^3 a^3+12 z a^3-2 a^3 z^{-1} -6 z^8 a^2+22 z^6 a^2-22 z^4 a^2+7 z^2 a^2-a^2-z^9 a-z^7 a+18 z^5 a-29 z^3 a+14 z a-2 a z^{-1} -2 z^8+9 z^6-12 z^4+5 z^2-z^7 a^{-1} +5 z^5 a^{-1} -8 z^3 a^{-1} +5 z a^{-1} - a^{-1} z^{-1} (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 \
6          11
4         1 -1
2        31 2
0       31  -2
-2      53   2
-4     44    0
-6    54     1
-8   35      2
-10  24       -2
-12 13        2
-14 2         -2
-161          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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See/edit the Link Page master template (intermediate).

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