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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 L11n258's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X19,22,20,9 X15,20,16,21 X21,16,22,17 X17,12,18,13 X2536 X9,1,10,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, -7, 8, -9, 5, -6, 7, -8, 6}
A Braid Representative
A Morse Link Presentation L11n258 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v w^3-4 u v w^2+5 u v w-3 u v-u w^3+2 u w^2-2 u w+u-v w^3+2 v w^2-2 v w+v+3 w^3-5 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial -3 q^{-6} +6 q^{-5} -10 q^{-4} +q^3+13 q^{-3} -2 q^2-12 q^{-2} +7 q+13 q^{-1} -9 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^6 z^{-2} -2 a^6-z^4 a^4+2 z^2 a^4+3 a^4 z^{-2} +6 a^4+z^6 a^2+2 z^4 a^2-2 a^2 z^{-2} -3 a^2-2 z^4-4 z^2- z^{-2} -3+z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} (db)
Kauffman polynomial 6 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +3 a^6 z^6+a^6 z^2-a^6 z^{-2} +a^6+7 a^5 z^7-17 a^5 z^5+32 a^5 z^3-27 a^5 z+8 a^5 z^{-1} +5 a^4 z^8-4 a^4 z^6-2 a^4 z^4+a^4 z^2-3 a^4 z^{-2} +5 a^4+a^3 z^9+14 a^3 z^7-43 a^3 z^5+51 a^3 z^3-34 a^3 z+10 a^3 z^{-1} +8 a^2 z^8-13 a^2 z^6+z^6 a^{-2} +a^2 z^4-4 z^4 a^{-2} -2 a^2 z^2+6 z^2 a^{-2} -2 a^2 z^{-2} + a^{-2} z^{-2} +4 a^2-4 a^{-2} +a z^9+9 a z^7+2 z^7 a^{-1} -30 a z^5-4 z^5 a^{-1} +25 a z^3-10 a z+4 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-5 z^6-z^4+4 z^2+ z^{-2} -3 (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 \
7         11
5        21-1
3       5  5
1      42  -2
-1     95   4
-3    67    1
-5   76     1
-7  36      3
-9 37       -4
-11 3        3
-133         -3
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-5 {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z} {\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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