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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 L11a18's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X18,14,19,13 X16,9,17,10 X8,17,9,18 X22,20,5,19 X20,15,21,16 X14,21,15,22 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, 4, -9, 8, -5, 6, -4, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a18 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^5}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+9 q^{7/2}-15 q^{5/2}+18 q^{3/2}-21 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -6 a^3 z+2 z a^{-3} -3 a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +9 a z^3-8 z^3 a^{-1} +10 a z-7 z a^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-3 a^3 z^9-9 a z^9-6 z^9 a^{-1} -3 a^4 z^8-12 a^2 z^8-13 z^8 a^{-2} -22 z^8-a^5 z^7+3 a^3 z^7+7 a z^7-11 z^7 a^{-1} -14 z^7 a^{-3} +11 a^4 z^6+47 a^2 z^6+18 z^6 a^{-2} -9 z^6 a^{-4} +63 z^6+4 a^5 z^5+16 a^3 z^5+41 a z^5+54 z^5 a^{-1} +21 z^5 a^{-3} -4 z^5 a^{-5} -14 a^4 z^4-51 a^2 z^4-5 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -50 z^4-6 a^5 z^3-31 a^3 z^3-61 a z^3-51 z^3 a^{-1} -14 z^3 a^{-3} +z^3 a^{-5} +7 a^4 z^2+20 a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +16 z^2+4 a^5 z+18 a^3 z+28 a z+19 z a^{-1} +5 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 a^{-1} 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 \
12           1-1
10          3 3
8         61 -5
6        93  6
4       96   -3
2      129    3
0     1011     1
-2    710      -3
-4   510       5
-6  27        -5
-8 15         4
-10 2          -2
-121           1
Integral Khovanov Homology

(db, data source)

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