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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 L11n45's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X17,1,18,4 X5,14,6,15 X3849 X9,18,10,19 X11,20,12,21 X13,22,14,5 X19,10,20,11 X21,12,22,13 X2,16,3,15
Gauss code {1, -11, -5, 3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, 11, -2, -3, 6, -9, 7, -10, 8}
A Braid Representative
A Morse Link Presentation L11n45 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right) \left(t(2)^2+t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z^5 a^9-5 z^3 a^9-7 z a^9-3 a^9 z^{-1} +z^7 a^7+7 z^5 a^7+18 z^3 a^7+20 z a^7+7 a^7 z^{-1} -2 z^5 a^5-10 z^3 a^5-13 z a^5-4 a^5 z^{-1} (db)
Kauffman polynomial a^{14} z^4-3 a^{14} z^2+a^{14}+a^{13} z^5-2 a^{13} z^3+a^{12} z^6-2 a^{12} z^4+a^{12} z^2+a^{11} z^7-3 a^{11} z^5+4 a^{11} z^3+a^{10} z^8-4 a^{10} z^6+6 a^{10} z^4+a^9 z^9-6 a^9 z^7+15 a^9 z^5-16 a^9 z^3+10 a^9 z-3 a^9 z^{-1} +2 a^8 z^8-11 a^8 z^6+24 a^8 z^4-23 a^8 z^2+7 a^8+a^7 z^9-7 a^7 z^7+22 a^7 z^5-36 a^7 z^3+26 a^7 z-7 a^7 z^{-1} +a^6 z^8-6 a^6 z^6+15 a^6 z^4-19 a^6 z^2+7 a^6+3 a^5 z^5-14 a^5 z^3+16 a^5 z-4 a^5 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 \
-4         22
-6        121
-8       2  2
-10      11  0
-12     32   1
-14    11    0
-16   23     -1
-18   1      1
-20 12       -1
-22          0
-241         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{2}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}^{2}

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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