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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 L11n137's Link Presentations]

Planar diagram presentation X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X13,20,14,21 X15,22,16,7 X4,20,5,19 X21,14,22,15
Gauss code {1, -4, 3, -10, 5, -6}, {6, -1, 7, -3, 2, -5, -8, 11, -9, -7, 4, -2, 10, 8, -11, 9}
A Braid Representative
A Morse Link Presentation L11n137 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-3 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+5 u v^3-7 u v^2+5 u v-u-v^3+3 v^2-3 v+1}{u v^2} (db)
Jones polynomial \frac{12}{q^{9/2}}-\frac{11}{q^{7/2}}+\frac{7}{q^{5/2}}-\frac{5}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{10}{q^{13/2}}-\frac{12}{q^{11/2}}+\frac{1}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^5+2 a^7 z^3-a^7 z^{-1} -a^5 z^7-4 a^5 z^5-4 a^5 z^3+2 a^5 z+3 a^5 z^{-1} +a^3 z^5+a^3 z^3-3 a^3 z-2 a^3 z^{-1} (db)
Kauffman polynomial a^{11} z^5-a^{11} z^3+4 a^{10} z^6-7 a^{10} z^4+2 a^{10} z^2+6 a^9 z^7-11 a^9 z^5+4 a^9 z^3+4 a^8 z^8-a^8 z^6-10 a^8 z^4+7 a^8 z^2-a^8+a^7 z^9+8 a^7 z^7-16 a^7 z^5+7 a^7 z^3-a^7 z+a^7 z^{-1} +5 a^6 z^8-2 a^6 z^6-8 a^6 z^4+9 a^6 z^2-3 a^6+a^5 z^9+2 a^5 z^7+a^5 z^5-2 a^5 z^3-4 a^5 z+3 a^5 z^{-1} +a^4 z^8+3 a^4 z^6-4 a^4 z^4+4 a^4 z^2-3 a^4+5 a^3 z^5-4 a^3 z^3-3 a^3 z+2 a^3 z^{-1} +a^2 z^4 (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        4 4
-4       42 -2
-6      73  4
-8     65   -1
-10    66    0
-12   46     2
-14  36      -3
-16 14       3
-18 3        -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
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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