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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 L11n207's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X17,22,18,9 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X13,18,14,19 X15,21,16,20
Gauss code {1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, -10, 4, -11, 8, -5, 10, -6, 11, -7, 5}
A Braid Representative
A Morse Link Presentation L11n207 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^3-u^3 v^2-2 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2+u v^3-4 u v^2+6 u v-2 u-v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial \frac{10}{q^{9/2}}-\frac{10}{q^{7/2}}+\frac{6}{q^{5/2}}-\frac{4}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{10}{q^{11/2}}+\frac{1}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+3 z^3 a^7+3 z a^7-z^7 a^5-5 z^5 a^5-9 z^3 a^5-5 z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-a^3 z^{-1} (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-3 z^2 a^{10}-4 z^7 a^9+6 z^5 a^9-z a^9-3 z^8 a^8+2 z^6 a^8+2 z^4 a^8+z^2 a^8-z^9 a^7-3 z^7 a^7+2 z^5 a^7+6 z^3 a^7-4 z a^7-4 z^8 a^6+5 z^6 a^6-6 z^4 a^6+5 z^2 a^6-z^9 a^5+z^7 a^5-9 z^5 a^5+13 z^3 a^5-5 z a^5-a^5 z^{-1} -z^8 a^4-3 z^4 a^4+2 z^2 a^4+a^4-4 z^5 a^3+5 z^3 a^3-z a^3-a^3 z^{-1} -z^4 a^2+z^2 a^2 (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 \
0         1-1
-2        3 3
-4       42 -2
-6      62  4
-8     44   0
-10    66    0
-12   45     1
-14  25      -3
-16 14       3
-18 2        -2
-201         1
Integral Khovanov Homology

(db, data source)

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