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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 L11a475's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,8,15,7 X20,16,21,15 X18,11,19,12 X12,17,13,18 X22,20,17,19 X16,22,5,21 X8,14,9,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, {10, -1, 3, -9, 11, -2, 5, -6, 9, -3, 4, -8}
A Braid Representative
A Morse Link Presentation L11a475 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(2)-1) (t(3)-1)^2 (t(3) t(1)-3 t(1)-3 t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial -q^5+4 q^4-9 q^3+15 q^2-19 q+21-20 q^{-1} +18 q^{-2} -11 q^{-3} +7 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 a^4 z^2-2 a^4 z^{-2} -z^2 a^{-4} -4 a^4+3 a^2 z^4+2 z^4 a^{-2} +4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +3 a^2-z^6-z^4-z^2 (db)
Kauffman polynomial a^2 z^{10}+z^{10}+3 a^3 z^9+8 a z^9+5 z^9 a^{-1} +3 a^4 z^8+11 a^2 z^8+9 z^8 a^{-2} +17 z^8+2 a^5 z^7-3 a z^7+7 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-3 a^4 z^6-27 a^2 z^6-11 z^6 a^{-2} +4 z^6 a^{-4} -38 z^6-4 a^5 z^5-7 a^3 z^5-18 a z^5-28 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-5 a^4 z^4+22 a^2 z^4+3 z^4 a^{-2} -5 z^4 a^{-4} +31 z^4+4 a^3 z^3+18 a z^3+22 z^3 a^{-1} +7 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+9 a^4 z^2-7 a^2 z^2-2 z^2 a^{-2} +2 z^2 a^{-4} -14 z^2+4 a^5 z+2 a^3 z-6 a z-6 z a^{-1} -2 z a^{-3} -4 a^6-6 a^4-a^2+ a^{-2} +3-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 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 \
11           1-1
9          3 3
7         61 -5
5        93  6
3       106   -4
1      119    2
-1     1112     1
-3    79      -2
-5   411       7
-7  37        -4
-9 16         5
-11 1          -1
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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