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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 L11a53's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X20,9,21,10 X8,19,9,20 X4,21,1,22 X14,6,15,5 X10,4,11,3 X16,12,17,11 X12,16,13,15 X22,14,5,13 X2,18,3,17
Gauss code {1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -9, 10, -6, 9, -8, 11, -2, 4, -3, 5, -10}
A Braid Representative
A Morse Link Presentation L11a53 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2+1\right) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{11/2}+4 q^{9/2}-7 q^{7/2}+11 q^{5/2}-16 q^{3/2}+17 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7+z^7 a^{-1} -a^3 z^5+4 a z^5+3 z^5 a^{-1} -z^5 a^{-3} -3 a^3 z^3+7 a z^3-2 z^3 a^{-3} -3 a^3 z+8 a z-6 z a^{-1} +z a^{-3} -a^3 z^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial a^6 z^4-a^6 z^2+z^7 a^{-5} +3 a^5 z^5-3 z^5 a^{-5} -2 a^5 z^3+2 z^3 a^{-5} +4 z^8 a^{-4} +6 a^4 z^6-15 z^6 a^{-4} -6 a^4 z^4+15 z^4 a^{-4} +4 a^4 z^2-z^2 a^{-4} -a^4- a^{-4} +5 z^9 a^{-3} +9 a^3 z^7-17 z^7 a^{-3} -15 a^3 z^5+16 z^5 a^{-3} +15 a^3 z^3-3 z^3 a^{-3} -6 a^3 z-2 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +2 z^{10} a^{-2} +8 a^2 z^8+3 z^8 a^{-2} -8 a^2 z^6-23 z^6 a^{-2} -6 a^2 z^4+18 z^4 a^{-2} +12 a^2 z^2+3 z^2 a^{-2} -4 a^2-4 a^{-2} +5 a z^9+10 z^9 a^{-1} +2 a z^7-25 z^7 a^{-1} -26 a z^5+11 z^5 a^{-1} +29 a z^3+7 z^3 a^{-1} -15 a z-11 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +2 z^{10}+7 z^8-22 z^6+4 z^4+11 z^2-7 (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 \
12           11
10          3 -3
8         41 3
6        73  -4
4       94   5
2      87    -1
0     109     1
-2    710      3
-4   58       -3
-6  27        5
-8 15         -4
-10 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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