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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 L11a35's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,8,17,7 X18,13,19,14 X14,17,15,18 X22,20,5,19 X20,12,21,11 X12,22,13,21 X8,16,9,15 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 7, -8, 4, -5, 9, -3, 5, -4, 6, -7, 8, -6}
A Braid Representative
A Morse Link Presentation L11a35 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-7 t(1) t(2)^3+9 t(2)^3+9 t(1) t(2)^2-7 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -3 q^{9/2}+\frac{3}{q^{9/2}}+6 q^{7/2}-\frac{7}{q^{7/2}}-10 q^{5/2}+\frac{10}{q^{5/2}}+14 q^{3/2}-\frac{14}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-16 \sqrt{q}+\frac{15}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} +z a^{-5} -3 a^3 z^3-2 z^3 a^{-3} -5 a^3 z-2 a^3 z^{-1} -z a^{-3} + a^{-3} z^{-1} +2 a z^5+z^5 a^{-1} +5 a z^3-z^3 a^{-1} +6 a z+3 a z^{-1} -4 z a^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-4 a^5 z^5+3 z^5 a^{-5} +6 a^5 z^3-3 z^3 a^{-5} -4 a^5 z+z a^{-5} +a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+5 z^6 a^{-4} +12 a^4 z^4-5 z^4 a^{-4} -4 a^4 z^2+3 z^2 a^{-4} - a^{-4} +3 a^3 z^9-5 a^3 z^7+6 z^7 a^{-3} -11 a^3 z^5-5 z^5 a^{-3} +22 a^3 z^3+z^3 a^{-3} -11 a^3 z-z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^{10}+8 a^2 z^8+6 z^8 a^{-2} -36 a^2 z^6-5 z^6 a^{-2} +38 a^2 z^4-5 z^4 a^{-2} -14 a^2 z^2+8 z^2 a^{-2} +2 a^2-2 a^{-2} +7 a z^9+4 z^9 a^{-1} -11 a z^7+z^7 a^{-1} -18 a z^5-19 z^5 a^{-1} +35 a z^3+23 z^3 a^{-1} -18 a z-13 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +z^{10}+11 z^8-35 z^6+27 z^4-6 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 \
12           1-1
10          2 2
8         41 -3
6        62  4
4       84   -4
2      86    2
0     89     1
-2    67      -1
-4   48       4
-6  36        -3
-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=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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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