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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 L11n130's Link Presentations]

Planar diagram presentation X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X9,16,10,17 X13,20,14,21 X15,22,16,7 X19,4,20,5 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 L11n130 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(v^2-v+1\right) (u v-u+1) (u v-v+1)}{u v^2} (db)
Jones polynomial \frac{1}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^5+3 a^7 z^3+2 a^7 z-a^7 z^{-1} -a^5 z^7-5 a^5 z^5-8 a^5 z^3-2 a^5 z+3 a^5 z^{-1} +a^3 z^5+2 a^3 z^3-a^3 z-2 a^3 z^{-1} (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+3 a^{10} z^6-7 a^{10} z^4+3 a^{10} z^2+4 a^9 z^7-9 a^9 z^5+5 a^9 z^3-a^9 z+3 a^8 z^8-5 a^8 z^6+2 a^8 z^4+a^8 z^2-a^8+a^7 z^9+2 a^7 z^7-5 a^7 z^5+3 a^7 z^3+a^7 z^{-1} +4 a^6 z^8-9 a^6 z^6+12 a^6 z^4-3 a^6 z^2-3 a^6+a^5 z^9-2 a^5 z^7+9 a^5 z^5-10 a^5 z^3+a^5 z+3 a^5 z^{-1} +a^4 z^8-a^4 z^6+4 a^4 z^4-2 a^4 z^2-3 a^4+4 a^3 z^5-6 a^3 z^3+2 a^3 z^{-1} +a^2 z^4-a^2 z^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       32 -1
-6      62  4
-8     44   0
-10    55    0
-12   34     1
-14  25      -3
-16 13       2
-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}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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