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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 L11n304's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -7, 6, -8, 5}, {11, -2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}
A Braid Representative
A Morse Link Presentation L11n304 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^2 w^3+u v^2 w^2+u v w^3-2 u v w^2+u v w+2 u w^2-2 u w+u-v^2 w^4+2 v^2 w^3-2 v^2 w^2-v w^3+2 v w^2-v w-w^2+w}{\sqrt{u} v w^2} (db)
Jones polynomial - q^{-7} +2 q^{-6} -4 q^{-5} +6 q^{-4} -6 q^{-3} +2 q^2+8 q^{-2} -3 q-6 q^{-1} +6 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^6-a^6 z^{-2} -3 a^6+3 z^4 a^4+12 z^2 a^4+4 a^4 z^{-2} +13 a^4-2 z^6 a^2-11 z^4 a^2-21 z^2 a^2-5 a^2 z^{-2} -17 a^2+2 z^4+7 z^2+2 z^{-2} +7 (db)
Kauffman polynomial a^7 z^7-5 a^7 z^5+8 a^7 z^3-5 a^7 z+a^7 z^{-1} +2 a^6 z^8-9 a^6 z^6+12 a^6 z^4-7 a^6 z^2-a^6 z^{-2} +4 a^6+a^5 z^9+a^5 z^7-20 a^5 z^5+35 a^5 z^3-21 a^5 z+5 a^5 z^{-1} +6 a^4 z^8-26 a^4 z^6+38 a^4 z^4-32 a^4 z^2-4 a^4 z^{-2} +17 a^4+a^3 z^9+4 a^3 z^7-28 a^3 z^5+44 a^3 z^3-33 a^3 z+9 a^3 z^{-1} +4 a^2 z^8-15 a^2 z^6+26 a^2 z^4-32 a^2 z^2+3 z^2 a^{-2} -5 a^2 z^{-2} +20 a^2-2 a^{-2} +4 a z^7-12 a z^5+z^5 a^{-1} +18 a z^3+z^3 a^{-1} -16 a z+z a^{-1} +5 a z^{-1} +2 z^6-4 z^2-2 z^{-2} +6 (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 \
5         22
3        32-1
1       3  3
-1      44  0
-3     42   2
-5    24    2
-7   44     0
-9  13      2
-11 13       -2
-13 1        1
-151         -1
Integral Khovanov Homology

(db, data source)

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

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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