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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 L11a137's Link Presentations]

Planar diagram presentation X6172 X2,11,3,12 X14,3,15,4 X12,5,13,6 X22,13,5,14 X4,21,1,22 X20,16,21,15 X16,8,17,7 X18,10,19,9 X8,18,9,17 X10,20,11,19
Gauss code {1, -2, 3, -6}, {4, -1, 8, -10, 9, -11, 2, -4, 5, -3, 7, -8, 10, -9, 11, -7, 6, -5}
A Braid Representative
A Morse Link Presentation L11a137 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^5-2 t(2)^5-5 t(1) t(2)^4+6 t(2)^4+6 t(1) t(2)^3-6 t(2)^3-6 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-5 t(2)-2 t(1)+2}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -4 q^{9/2}+\frac{3}{q^{9/2}}+8 q^{7/2}-\frac{6}{q^{7/2}}-12 q^{5/2}+\frac{9}{q^{5/2}}+16 q^{3/2}-\frac{15}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-17 \sqrt{q}+\frac{16}{\sqrt{q}} (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^3-6 a z^3-z^3 a^{-1} +2 z^3 a^{-3} +2 a^3 z-6 a z+2 z a^{-1} +2 a^3 z^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-4 a^3 z^9-10 a z^9-6 z^9 a^{-1} -3 a^4 z^8-9 z^8 a^{-2} -6 z^8-a^5 z^7+15 a^3 z^7+33 a z^7+7 z^7 a^{-1} -10 z^7 a^{-3} +12 a^4 z^6+17 a^2 z^6+11 z^6 a^{-2} -8 z^6 a^{-4} +24 z^6+4 a^5 z^5-18 a^3 z^5-39 a z^5-z^5 a^{-1} +12 z^5 a^{-3} -4 z^5 a^{-5} -13 a^4 z^4-23 a^2 z^4+8 z^4 a^{-4} -z^4 a^{-6} -19 z^4-4 a^5 z^3+11 a^3 z^3+24 a z^3+4 z^3 a^{-1} -3 z^3 a^{-3} +2 z^3 a^{-5} +3 a^4 z^2+10 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +7 z^2+a^5 z-5 a^3 z-8 a z-2 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 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 \
12           1-1
10          3 3
8         51 -4
6        73  4
4       95   -4
2      87    1
0     910     1
-2    67      -1
-4   39       6
-6  36        -3
-8 14         3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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

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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