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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 L11a349's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X14,6,15,5 X18,8,19,7 X20,9,21,10 X10,11,1,12 X6,14,7,13 X4,18,5,17 X22,15,11,16 X2,19,3,20 X16,21,17,22
Gauss code {1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -11, 8, -4, 10, -5, 11, -9}
A Braid Representative
A Morse Link Presentation L11a349 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-3 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)+t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+10 q^{5/2}-18 q^{3/2}+23 \sqrt{q}-\frac{28}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z-2 a^3 z^5-3 a^3 z^3+z^3 a^{-3} -a^3 z+z a^{-3} +a z^7+2 a z^5-2 z^5 a^{-1} +a z^3-3 z^3 a^{-1} -z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -3 a^2 z^{10}-3 z^{10}-10 a^3 z^9-18 a z^9-8 z^9 a^{-1} -13 a^4 z^8-22 a^2 z^8-8 z^8 a^{-2} -17 z^8-9 a^5 z^7+5 a^3 z^7+26 a z^7+8 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+21 a^4 z^6+58 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +50 z^6-a^7 z^5+12 a^5 z^5+13 a^3 z^5+8 z^5 a^{-1} +8 z^5 a^{-3} +4 a^6 z^4-17 a^4 z^4-46 a^2 z^4-11 z^4 a^{-2} +2 z^4 a^{-4} -38 z^4+a^7 z^3-8 a^5 z^3-12 a^3 z^3-5 a z^3-8 z^3 a^{-1} -6 z^3 a^{-3} -a^6 z^2+6 a^4 z^2+14 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +11 z^2+2 a^5 z+2 a^3 z+2 z a^{-1} +2 z a^{-3} +1-a z^{-1} - 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 \
10           1-1
8          3 3
6         71 -6
4        113  8
2       127   -5
0      1611    5
-2     1314     1
-4    1114      -3
-6   713       6
-8  311        -8
-10 17         6
-12 3          -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}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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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