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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 L11a104's Link Presentations]

Planar diagram presentation X6172 X2,9,3,10 X12,3,13,4 X20,11,21,12 X18,6,19,5 X22,20,5,19 X10,21,11,22 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15
Gauss code {1, -2, 3, -8}, {5, -1, 10, -11, 2, -7, 4, -3, 9, -10, 11, -9, 8, -5, 6, -4, 7, -6}
A Braid Representative
A Morse Link Presentation L11a104 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-6 u v^4+12 u v^3-12 u v^2+4 u v+4 v^4-12 v^3+12 v^2-6 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{9/2}-5 q^{7/2}+10 q^{5/2}-15 q^{3/2}+20 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-6 a^3 z^3+9 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-9 a^3 z+8 a z-3 z a^{-1} +2 a^5 z^{-1} -4 a^3 z^{-1} +3 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-12 a z^9-7 z^9 a^{-1} -7 a^4 z^8-12 a^2 z^8-9 z^8 a^{-2} -14 z^8-6 a^5 z^7-4 a^3 z^7+15 a z^7+8 z^7 a^{-1} -5 z^7 a^{-3} -3 a^6 z^6+8 a^4 z^6+27 a^2 z^6+21 z^6 a^{-2} -z^6 a^{-4} +38 z^6-a^7 z^5+10 a^5 z^5+24 a^3 z^5+11 a z^5+8 z^5 a^{-1} +10 z^5 a^{-3} +4 a^6 z^4-3 a^4 z^4-14 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -19 z^4+2 a^7 z^3-10 a^5 z^3-32 a^3 z^3-24 a z^3-7 z^3 a^{-1} -3 z^3 a^{-3} -a^6 z^2-2 a^4 z^2-4 a^2 z^2-z^2 a^{-2} -4 z^2-a^7 z+7 a^5 z+18 a^3 z+13 a z+2 z a^{-1} -z a^{-3} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 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          4 4
6         61 -5
4        94  5
2       116   -5
0      129    3
-2     1112     1
-4    811      -3
-6   511       6
-8  38        -5
-10 16         5
-12 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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

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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