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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 L11a244's Link Presentations]

Planar diagram presentation X8192 X10,8,11,7 X12,3,13,4 X18,5,19,6 X16,12,17,11 X14,20,15,19 X20,14,21,13 X2,15,3,16 X22,18,7,17 X4,21,5,22 X6,9,1,10
Gauss code {1, -8, 3, -10, 4, -11}, {2, -1, 11, -2, 5, -3, 7, -6, 8, -5, 9, -4, 6, -7, 10, -9}
A Braid Representative
A Morse Link Presentation L11a244 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4-5 t(1)^2 t(2)^3+11 t(1) t(2)^3-6 t(2)^3+9 t(1)^2 t(2)^2-19 t(1) t(2)^2+9 t(2)^2-6 t(1)^2 t(2)+11 t(1) t(2)-5 t(2)-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{9/2}-\frac{11}{q^{9/2}}-5 q^{7/2}+\frac{17}{q^{7/2}}+11 q^{5/2}-\frac{25}{q^{5/2}}-18 q^{3/2}+\frac{28}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+25 \sqrt{q}-\frac{28}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z+2 a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -6 a^3 z-3 a^3 z^{-1} +a z^7+3 a z^5-2 z^5 a^{-1} +6 a z^3-3 z^3 a^{-1} +4 a z-2 z a^{-1} +a z^{-1} (db)
Kauffman polynomial -3 a^2 z^{10}-3 z^{10}-10 a^3 z^9-19 a z^9-9 z^9 a^{-1} -13 a^4 z^8-24 a^2 z^8-10 z^8 a^{-2} -21 z^8-10 a^5 z^7+3 a^3 z^7+26 a z^7+8 z^7 a^{-1} -5 z^7 a^{-3} -4 a^6 z^6+18 a^4 z^6+62 a^2 z^6+21 z^6 a^{-2} -z^6 a^{-4} +62 z^6-a^7 z^5+16 a^5 z^5+17 a^3 z^5+5 a z^5+14 z^5 a^{-1} +9 z^5 a^{-3} +3 a^6 z^4-8 a^4 z^4-46 a^2 z^4-13 z^4 a^{-2} +z^4 a^{-4} -49 z^4+a^7 z^3-15 a^5 z^3-16 a^3 z^3-9 a z^3-13 z^3 a^{-1} -4 z^3 a^{-3} -3 a^4 z^2+9 a^2 z^2+2 z^2 a^{-2} +14 z^2+8 a^5 z+7 a^3 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 \
10           1-1
8          4 4
6         71 -6
4        114  7
2       147   -7
0      1411    3
-2     1515     0
-4    1013      -3
-6   715       8
-8  410        -6
-10  7         7
-1214          -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} {\mathbb Z}
r=-5 {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\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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