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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 L10a51's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X12,5,13,6 X16,12,17,11 X6,18,1,17 X14,19,15,20 X18,13,19,14
Gauss code {1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -10, 9, -3}
A Braid Representative
A Morse Link Presentation L10a51 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(t(1) t(2)^2-t(2)^2-2 t(1) t(2)+t(2)+t(1)-1\right) \left(t(1) t(2)^2-t(2)^2-t(1) t(2)+2 t(2)+t(1)-1\right)}{t(1) t(2)^2} (db)
Jones polynomial -\frac{8}{q^{9/2}}-q^{7/2}+\frac{12}{q^{7/2}}+4 q^{5/2}-\frac{16}{q^{5/2}}-8 q^{3/2}+\frac{16}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+12 \sqrt{q}-\frac{16}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z-2 a^3 z^5-5 a^3 z^3-3 a^3 z+a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +6 a z^3-2 z^3 a^{-1} +2 a z-z a^{-1} -a z^{-1} (db)
Kauffman polynomial a^7 z^5-a^7 z^3+4 a^6 z^6-6 a^6 z^4+2 a^6 z^2+7 a^5 z^7-12 a^5 z^5+7 a^5 z^3-2 a^5 z+6 a^4 z^8-4 a^4 z^6-6 a^4 z^4+4 a^4 z^2+2 a^3 z^9+13 a^3 z^7-33 a^3 z^5+z^5 a^{-3} +24 a^3 z^3-z^3 a^{-3} -5 a^3 z-a^3 z^{-1} +12 a^2 z^8-16 a^2 z^6+4 z^6 a^{-2} -6 z^4 a^{-2} +4 a^2 z^2+2 z^2 a^{-2} +a^2+2 a z^9+13 a z^7+7 z^7 a^{-1} -33 a z^5-12 z^5 a^{-1} +24 a z^3+7 z^3 a^{-1} -5 a z-2 z a^{-1} -a z^{-1} +6 z^8-4 z^6-6 z^4+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 \
8          11
6         3 -3
4        51 4
2       73  -4
0      95   4
-2     88    0
-4    88     0
-6   59      4
-8  37       -4
-10 15        4
-12 3         -3
-141          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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