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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a2 at Knotilus!

Link Presentations

[edit Notes on L11a2's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X18,11,19,12 X20,14,21,13 X22,19,5,20 X12,22,13,21 X2,16,3,15
Gauss code {1, -11, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -10, 8, -4, 11, -2, 3, -7, 9, -8, 10, -9}
A Braid Representative
A Morse Link Presentation L11a2 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{9/2}-\frac{11}{q^{9/2}}-5 q^{7/2}+\frac{17}{q^{7/2}}+10 q^{5/2}-\frac{24}{q^{5/2}}-17 q^{3/2}+\frac{27}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{5}{q^{11/2}}+23 \sqrt{q}-\frac{27}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+3 a z^5-2 z^5 a^{-1} +a^5 z^3-3 a^3 z^3+5 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -a^3 z+a z+a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -13 a^4 z^8-26 a^2 z^8-9 z^8 a^{-2} -22 z^8-11 a^5 z^7-8 a^3 z^7+9 a z^7+z^7 a^{-1} -5 z^7 a^{-3} -5 a^6 z^6+16 a^4 z^6+58 a^2 z^6+19 z^6 a^{-2} -z^6 a^{-4} +57 z^6-a^7 z^5+15 a^5 z^5+35 a^3 z^5+32 a z^5+23 z^5 a^{-1} +10 z^5 a^{-3} +4 a^6 z^4-5 a^4 z^4-35 a^2 z^4-12 z^4 a^{-2} +z^4 a^{-4} -39 z^4-6 a^5 z^3-21 a^3 z^3-27 a z^3-17 z^3 a^{-1} -5 z^3 a^{-3} +a^4 z^2+6 a^2 z^2+2 z^2 a^{-2} +7 z^2+a^3 z-2 z a^{-1} -z a^{-3} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} + a^{-3} 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         61 -5
4        114  7
2       126   -6
0      1511    4
-2     1414     0
-4    1013      -3
-6   714       7
-8  410        -6
-10 17         6
-12 4          -4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
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}^{14}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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