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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 L10a93's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X4,20,5,19 X14,5,15,6 X18,13,19,14 X16,7,17,8 X6,15,7,16 X8,17,1,18
Gauss code {1, -4, 2, -5, 6, -9, 8, -10}, {4, -1, 3, -2, 7, -6, 9, -8, 10, -7, 5, -3}
A Braid Representative
A Morse Link Presentation L10a93 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^3-2 u^3 v^2+u^3 v-2 u^2 v^3+5 u^2 v^2-3 u^2 v+u^2+u v^3-3 u v^2+5 u v-2 u+v^2-2 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7+2 z^5 a^5+7 z^3 a^5+6 z a^5+a^5 z^{-1} -z^7 a^3-5 z^5 a^3-9 z^3 a^3-7 z a^3-a^3 z^{-1} +z^5 a+3 z^3 a+z a (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+4 z^3 a^9-z a^9-4 z^6 a^8+4 z^4 a^8-4 z^7 a^7+4 z^5 a^7-z^3 a^7-3 z^8 a^6+3 z^6 a^6-2 z^4 a^6+z^2 a^6-z^9 a^5-5 z^7 a^5+18 z^5 a^5-21 z^3 a^5+9 z a^5-a^5 z^{-1} -6 z^8 a^4+17 z^6 a^4-16 z^4 a^4+4 z^2 a^4+a^4-z^9 a^3-4 z^7 a^3+22 z^5 a^3-26 z^3 a^3+10 z a^3-a^3 z^{-1} -3 z^8 a^2+9 z^6 a^2-6 z^4 a^2+z^2 a^2-3 z^7 a+11 z^5 a-10 z^3 a+2 z a-z^6+3 z^4-z^2 (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 \
4          1-1
2         2 2
0        21 -1
-2       52  3
-4      53   -2
-6     54    1
-8    45     1
-10   45      -1
-12  25       3
-14 13        -2
-16 2         2
-181          -1
Integral Khovanov Homology

(db, data source)

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