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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 L10a85's Link Presentations]

Planar diagram presentation X8192 X16,5,17,6 X18,10,19,9 X10,20,11,19 X14,18,15,17 X2,11,3,12 X12,3,13,4 X4758 X20,14,7,13 X6,15,1,16
Gauss code {1, -6, 7, -8, 2, -10}, {8, -1, 3, -4, 6, -7, 9, -5, 10, -2, 5, -3, 4, -9}
A Braid Representative
A Morse Link Presentation L10a85 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4-3 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+4 t(1)^2 t(2)^2-9 t(1) t(2)^2+4 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-3 t(2)-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial -\frac{8}{q^{9/2}}-q^{7/2}+\frac{11}{q^{7/2}}+4 q^{5/2}-\frac{15}{q^{5/2}}-8 q^{3/2}+\frac{16}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+12 \sqrt{q}-\frac{15}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+4 a z^5-z^5 a^{-1} +a^5 z^3-6 a^3 z^3+6 a z^3-2 z^3 a^{-1} +2 a^5 z-7 a^3 z+3 a z-z a^{-1} +2 a^5 z^{-1} -3 a^3 z^{-1} +a z^{-1} (db)
Kauffman polynomial -2 a^3 z^9-2 a z^9-5 a^4 z^8-11 a^2 z^8-6 z^8-6 a^5 z^7-10 a^3 z^7-11 a z^7-7 z^7 a^{-1} -3 a^6 z^6+2 a^4 z^6+14 a^2 z^6-4 z^6 a^{-2} +5 z^6-a^7 z^5+12 a^5 z^5+26 a^3 z^5+26 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +4 a^6 z^4+9 a^4 z^4+3 a^2 z^4+6 z^4 a^{-2} +4 z^4+2 a^7 z^3-13 a^5 z^3-23 a^3 z^3-15 a z^3-6 z^3 a^{-1} +z^3 a^{-3} -a^6 z^2-10 a^4 z^2-10 a^2 z^2-2 z^2 a^{-2} -3 z^2-a^7 z+9 a^5 z+12 a^3 z+3 a z+z a^{-1} +3 a^4+3 a^2+1-2 a^5 z^{-1} -3 a^3 z^{-1} -a 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 \
8          11
6         3 -3
4        51 4
2       73  -4
0      85   3
-2     98    -1
-4    67     -1
-6   59      4
-8  36       -3
-10  5        5
-1213         -2
-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} {\mathbb Z}
r=-5 {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
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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