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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a20 at Knotilus!

Link Presentations

[edit Notes on L10a20's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X20,15,5,16 X16,7,17,8 X12,18,13,17 X14,10,15,9 X18,14,19,13 X8,19,9,20 X2536 X4,12,1,11
Gauss code {1, -9, 2, -10}, {9, -1, 4, -8, 6, -2, 10, -5, 7, -6, 3, -4, 5, -7, 8, -3}
A Braid Representative
A Morse Link Presentation L10a20 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^3 \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+9 q^{7/2}-13 q^{5/2}+15 q^{3/2}-17 \sqrt{q}+\frac{14}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-3} -a^3 z^3+2 z^3 a^{-3} -2 a^3 z+2 z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +6 a z^3-7 z^3 a^{-1} +7 a z-7 z a^{-1} +4 a z^{-1} -4 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} +4 z^5 a^{-5} -z^3 a^{-5} +a^4 z^6+9 z^6 a^{-4} -3 a^4 z^4-9 z^4 a^{-4} +3 a^4 z^2+4 z^2 a^{-4} -a^4- a^{-4} +3 a^3 z^7+12 z^7 a^{-3} -8 a^3 z^5-18 z^5 a^{-3} +8 a^3 z^3+11 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +4 a^2 z^8+8 z^8 a^{-2} -6 a^2 z^6-3 z^6 a^{-2} -4 a^2 z^4-14 z^4 a^{-2} +10 a^2 z^2+12 z^2 a^{-2} -4 a^2-4 a^{-2} +2 a z^9+2 z^9 a^{-1} +8 a z^7+17 z^7 a^{-1} -31 a z^5-45 z^5 a^{-1} +29 a z^3+33 z^3 a^{-1} -13 a z-13 z a^{-1} +4 a z^{-1} +4 a^{-1} z^{-1} +12 z^8-19 z^6-5 z^4+15 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          1-1
10         3 3
8        61 -5
6       73  4
4      86   -2
2     97    2
0    710     3
-2   57      -2
-4  27       5
-6 15        -4
-8 2         2
-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}^{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}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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

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

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