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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 L11n110's Link Presentations]

Planar diagram presentation X6172 X3,12,4,13 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X11,17,12,16 X17,15,18,14 X15,11,16,10 X2536 X13,4,14,1
Gauss code {1, -10, -2, 11}, {10, -1, -3, 6, -5, 9, -7, 2, -11, 8, -9, 7, -8, 3, -4, 5, -6, 4}
A Braid Representative
A Morse Link Presentation L11n110 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-4 u v^4+7 u v^3-6 u v^2+u v+v^4-6 v^3+7 v^2-4 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{13}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+5 a z^5-2 z^5 a^{-1} -4 a^3 z^3+11 a z^3-7 z^3 a^{-1} +z^3 a^{-3} +a^5 z-8 a^3 z+12 a z-9 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -5 a^3 z^{-1} +6 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial 6 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +3 a^4 z^6+z^6 a^{-4} -3 z^4 a^{-4} +a^4 z^2+3 z^2 a^{-4} -a^4- a^{-4} +7 a^3 z^7+3 z^7 a^{-3} -17 a^3 z^5-9 z^5 a^{-3} +29 a^3 z^3+9 z^3 a^{-3} -21 a^3 z-4 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +5 a^2 z^8+3 z^8 a^{-2} -5 a^2 z^6-3 z^6 a^{-2} -9 z^4 a^{-2} +3 a^2 z^2+11 z^2 a^{-2} -a^2-3 a^{-2} +a z^9+z^9 a^{-1} +14 a z^7+10 z^7 a^{-1} -44 a z^5-36 z^5 a^{-1} +51 a z^3+37 z^3 a^{-1} -29 a z-19 z a^{-1} +6 a z^{-1} +4 a^{-1} z^{-1} +8 z^8-12 z^6-6 z^4+10 z^2-3 (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        2 2
6       41 -3
4      62  4
2     64   -2
0    76    1
-2   67     1
-4  46      -2
-6 26       4
-814        -3
-103         3
Integral Khovanov Homology

(db, data source)

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