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

Visit L11n318 at Knotilus!

Link Presentations

[edit Notes on L11n318's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,5,15,6 X20,11,21,12 X22,17,11,18 X16,21,17,22 X10,13,5,14 X19,8,20,9 X7,18,8,19 X2,9,3,10 X4,16,1,15
Gauss code {1, -10, 2, -11}, {3, -1, -9, 8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, 9, -8, -4, 6, -5}
A Braid Representative
A Morse Link Presentation L11n318 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^4-3 u v w^3+5 u v w^2-3 u v w+u w^3-2 u w^2+2 u w-2 v^2 w^3+2 v^2 w^2-v^2 w+3 v w^3-5 v w^2+3 v w-v-w^3+2 w^2-w}{\sqrt{u} v w^2} (db)
Jones polynomial - q^{-11} +4 q^{-10} -8 q^{-9} +11 q^{-8} -14 q^{-7} +15 q^{-6} -12 q^{-5} +11 q^{-4} -5 q^{-3} +3 q^{-2} (db)
Signature -4 (db)
HOMFLY-PT polynomial -z^2 a^{10}-a^{10} z^{-2} -a^{10}+3 z^4 a^8+8 z^2 a^8+4 a^8 z^{-2} +8 a^8-2 z^6 a^6-9 z^4 a^6-17 z^2 a^6-5 a^6 z^{-2} -15 a^6+3 z^4 a^4+9 z^2 a^4+2 a^4 z^{-2} +8 a^4 (db)
Kauffman polynomial a^{13} z^5-a^{13} z^3+4 a^{12} z^6-6 a^{12} z^4+a^{12} z^2+7 a^{11} z^7-13 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +6 a^{10} z^8-8 a^{10} z^6+2 a^{10} z^4-3 a^{10} z^2-a^{10} z^{-2} +2 a^{10}+2 a^9 z^9+9 a^9 z^7-28 a^9 z^5+27 a^9 z^3-13 a^9 z+5 a^9 z^{-1} +11 a^8 z^8-27 a^8 z^6+35 a^8 z^4-27 a^8 z^2-4 a^8 z^{-2} +13 a^8+2 a^7 z^9+5 a^7 z^7-20 a^7 z^5+33 a^7 z^3-27 a^7 z+9 a^7 z^{-1} +5 a^6 z^8-15 a^6 z^6+33 a^6 z^4-37 a^6 z^2-5 a^6 z^{-2} +20 a^6+3 a^5 z^7-6 a^5 z^5+13 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +6 a^4 z^4-14 a^4 z^2-2 a^4 z^{-2} +10 a^4 (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 \
-3         33
-5        42-2
-7       71 6
-9      65  -1
-11     96   3
-13    56    1
-15   69     -3
-17  36      3
-19 15       -4
-21 3        3
-231         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}

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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