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

Visit L11n80 at Knotilus!

Link Presentations

[edit Notes on L11n80's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X19,1,20,4 X5,14,6,15 X3,10,4,11 X11,20,12,21 X13,22,14,5 X21,12,22,13 X2,16,3,15
Gauss code {1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -8, 10, -9, 6, 11, -2, 4, -3, -5, 8, -10, 9}
A Braid Representative
A Morse Link Presentation L11n80 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^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{2}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{7}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z^5 a^9-4 z^3 a^9-6 z a^9-3 a^9 z^{-1} +z^7 a^7+6 z^5 a^7+15 z^3 a^7+18 z a^7+7 a^7 z^{-1} -2 z^5 a^5-9 z^3 a^5-12 z a^5-4 a^5 z^{-1} (db)
Kauffman polynomial a^{14} z^4-2 a^{14} z^2+a^{14}+2 a^{13} z^5-2 a^{13} z^3+3 a^{12} z^6-3 a^{12} z^4+a^{12} z^2+3 a^{11} z^7-3 a^{11} z^5+2 a^{11} z^3+2 a^{10} z^8-a^{10} z^6+a^{10} z^2+a^9 z^9-a^9 z^7+6 a^9 z^5-11 a^9 z^3+8 a^9 z-3 a^9 z^{-1} +3 a^8 z^8-8 a^8 z^6+15 a^8 z^4-19 a^8 z^2+7 a^8+a^7 z^9-4 a^7 z^7+14 a^7 z^5-27 a^7 z^3+22 a^7 z-7 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+11 a^6 z^4-17 a^6 z^2+7 a^6+3 a^5 z^5-12 a^5 z^3+14 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        220
-8       4  4
-10      32  -1
-12     54   1
-14    33    0
-16   45     -1
-18  13      2
-20 14       -3
-22 1        1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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