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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 L11a389's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X20,12,21,11 X22,17,9,18 X18,21,19,22 X16,14,17,13 X8,16,5,15 X14,8,15,7 X12,20,13,19 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 8, -7}, {11, -2, 3, -9, 6, -8, 7, -6, 4, -5, 9, -3, 5, -4}
A Braid Representative
A Morse Link Presentation L11a389 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-t(1) t(3)^3+t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+6 t(1) t(3)^2-5 t(1) t(2) t(3)^2+8 t(2) t(3)^2-5 t(3)^2-8 t(1) t(3)+5 t(1) t(2) t(3)-6 t(2) t(3)+5 t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial -q^5+4 q^4-8 q^3+14 q^2-17 q+19-18 q^{-1} +16 q^{-2} -10 q^{-3} +6 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+3 z^4 a^2+6 z^2 a^2+4 a^2 z^{-2} +8 a^2-z^6-2 z^4-5 z^2-3 z^{-2} -6+2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^2 a^{-4} (db)
Kauffman polynomial a^2 z^{10}+z^{10}+2 a^3 z^9+6 a z^9+4 z^9 a^{-1} +2 a^4 z^8+5 a^2 z^8+7 z^8 a^{-2} +10 z^8+2 a^5 z^7+3 a^3 z^7-a z^7+5 z^7 a^{-1} +7 z^7 a^{-3} +a^6 z^6+2 a^4 z^6-a^2 z^6-5 z^6 a^{-2} +4 z^6 a^{-4} -11 z^6-5 a^5 z^5-12 a^3 z^5-11 a z^5-15 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-18 a^4 z^4-26 a^2 z^4-7 z^4 a^{-2} -6 z^4 a^{-4} -13 z^4+3 a^5 z^3+8 a^3 z^3+5 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+25 a^4 z^2+38 a^2 z^2+8 z^2 a^{-2} +3 z^2 a^{-4} +24 z^2+a^5 z+a^3 z+a z+z a^{-1} -4 a^6-14 a^4-21 a^2-4 a^{-2} -14-a^5 z^{-1} -a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +a^6 z^{-2} +3 a^4 z^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} +3 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         51 -4
5        93  6
3       85   -3
1      119    2
-1     1011     1
-3    68      -2
-5   410       6
-7  26        -4
-9 15         4
-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}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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See/edit the Link Page master template (intermediate).

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