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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 L11a69's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,10,15,9 X22,20,5,19 X20,11,21,12 X10,21,11,22 X18,15,19,16 X16,7,17,8 X8,17,9,18 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 5, -2, 11, -3, 7, -8, 9, -7, 4, -5, 6, -4}
A Braid Representative
A Morse Link Presentation L11a69 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^4-4 v^3+7 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+14 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-8 a^3 z-3 a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +8 a z^3-2 z^3 a^{-1} +6 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-9 a^3 z^9-5 a z^9-5 a^6 z^8-16 a^4 z^8-20 a^2 z^8-9 z^8-3 a^7 z^7-2 a^5 z^7-9 a z^7-8 z^7 a^{-1} -a^8 z^6+9 a^6 z^6+41 a^4 z^6+47 a^2 z^6-4 z^6 a^{-2} +12 z^6+7 a^7 z^5+21 a^5 z^5+36 a^3 z^5+36 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-3 a^6 z^4-32 a^4 z^4-37 a^2 z^4+5 z^4 a^{-2} -6 z^4-6 a^7 z^3-22 a^5 z^3-41 a^3 z^3-35 a z^3-9 z^3 a^{-1} +z^3 a^{-3} -3 a^8 z^2-2 a^6 z^2+9 a^4 z^2+11 a^2 z^2-z^2 a^{-2} +2 z^2+3 a^7 z+12 a^5 z+18 a^3 z+13 a z+4 z a^{-1} +a^8+2 a^6-2 a^2-a^7 z^{-1} -3 a^5 z^{-1} -3 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1} (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 \
8           11
6          3 -3
4         61 5
2        83  -5
0       126   6
-2      1210    -2
-4     1010     0
-6    812      4
-8   610       -4
-10  28        6
-12 16         -5
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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