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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 L10a53's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X14,20,15,19 X2,18,3,17
Gauss code {1, -10, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, 4, -9, 7, -8, 10, -2, 9, -4}
A Braid Representative
A Morse Link Presentation L10a53 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(u v^2-2 u v+u+2 v-1\right) \left(u v^2-2 u v-v^2+2 v-1\right)}{u v^2} (db)
Jones polynomial q^{9/2}-\frac{4}{q^{9/2}}-4 q^{7/2}+\frac{8}{q^{7/2}}+8 q^{5/2}-\frac{13}{q^{5/2}}-12 q^{3/2}+\frac{15}{q^{3/2}}+\frac{1}{q^{11/2}}+15 \sqrt{q}-\frac{17}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-2 a^3 z^3+z^3 a^{-3} -a^3 z+z a^{-3} +a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +6 a z^3-5 z^3 a^{-1} +2 a z-3 z a^{-1} -a z^{-1} (db)
Kauffman polynomial a^6 z^4+4 a^5 z^5-2 a^5 z^3+8 a^4 z^6+z^6 a^{-4} -7 a^4 z^4-2 z^4 a^{-4} +a^4 z^2+z^2 a^{-4} +11 a^3 z^7+4 z^7 a^{-3} -16 a^3 z^5-10 z^5 a^{-3} +9 a^3 z^3+7 z^3 a^{-3} -a^3 z-2 z a^{-3} -a^3 z^{-1} +9 a^2 z^8+6 z^8 a^{-2} -11 a^2 z^6-14 z^6 a^{-2} +2 a^2 z^4+7 z^4 a^{-2} +a^2+3 a z^9+3 z^9 a^{-1} +12 a z^7+5 z^7 a^{-1} -39 a z^5-29 z^5 a^{-1} +29 a z^3+25 z^3 a^{-1} -5 a z-6 z a^{-1} -a z^{-1} +15 z^8-34 z^6+19 z^4-2 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 \
10          1-1
8         3 3
6        51 -4
4       73  4
2      85   -3
0     97    2
-2    79     2
-4   68      -2
-6  38       5
-8 15        -4
-10 3         3
-121          -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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