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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 L11a13's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X12,6,13,5 X8493 X16,10,17,9 X18,13,19,14 X14,17,15,18 X10,16,11,15 X22,12,5,11 X2,20,3,19
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 6, -9, 10, -4, 7, -8, 9, -6, 8, -7, 11, -2, 3, -10}
A Braid Representative
A Morse Link Presentation L11a13 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-9 q^{9/2}+16 q^{7/2}-22 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+4 a z^3-5 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a^3 z+3 a z-4 z a^{-1} +3 z a^{-3} -z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10}-6 a z^9-13 z^9 a^{-1} -7 z^9 a^{-3} -7 a^2 z^8-19 z^8 a^{-2} -10 z^8 a^{-4} -16 z^8-4 a^3 z^7+2 a z^7+11 z^7 a^{-1} -3 z^7 a^{-3} -8 z^7 a^{-5} -a^4 z^6+14 a^2 z^6+43 z^6 a^{-2} +12 z^6 a^{-4} -4 z^6 a^{-6} +42 z^6+9 a^3 z^5+18 a z^5+20 z^5 a^{-1} +23 z^5 a^{-3} +11 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-8 a^2 z^4-27 z^4 a^{-2} -4 z^4 a^{-4} +5 z^4 a^{-6} -28 z^4-7 a^3 z^3-18 a z^3-23 z^3 a^{-1} -20 z^3 a^{-3} -7 z^3 a^{-5} +z^3 a^{-7} -a^4 z^2+a^2 z^2+5 z^2 a^{-2} -z^2 a^{-4} -2 z^2 a^{-6} +6 z^2+2 a^3 z+6 a z+8 z a^{-1} +6 z a^{-3} +2 z a^{-5} +1-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 \
14           11
12          3 -3
10         61 5
8        103  -7
6       126   6
4      1210    -2
2     1312     1
0    1014      4
-2   611       -5
-4  310        7
-6 16         -5
-8 3          3
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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

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

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