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

Visit L11n228 at Knotilus!

Link Presentations

[edit Notes on L11n228's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X17,9,18,22 X9,21,10,20 X6,13,7,14 X14,7,15,8 X8,15,1,16 X19,5,20,4 X5,19,6,18 X21,17,22,16
Gauss code {1, -2, 3, 9, -10, -6, 7, -8}, {-5, -1, 2, -3, 6, -7, 8, 11, -4, 10, -9, 5, -11, 4}
A Braid Representative
A Morse Link Presentation L11n228 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^3-u^3 v^2+u^2 v^3-u^2 v^2+2 u^2 v-u^2-u v^3+2 u v^2-u v+u-v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial -\frac{3}{q^{9/2}}+q^{7/2}+\frac{3}{q^{7/2}}-q^{5/2}-\frac{3}{q^{5/2}}+q^{3/2}+\frac{2}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}-\sqrt{q}-\frac{2}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z+2 a^3 z^3+4 a^3 z+2 a^3 z^{-1} -a z^7-7 a z^5+z^5 a^{-1} -15 a z^3+5 z^3 a^{-1} -13 a z+6 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -a^2 z^8-z^8 a^{-2} -2 z^8-a^5 z^7+8 a z^7+7 z^7 a^{-1} -2 a^6 z^6-2 a^4 z^6+8 a^2 z^6+7 z^6 a^{-2} +15 z^6-a^7 z^5+a^5 z^5-2 a^3 z^5-19 a z^5-15 z^5 a^{-1} +6 a^6 z^4+4 a^4 z^4-18 a^2 z^4-15 z^4 a^{-2} -31 z^4+3 a^7 z^3+3 a^5 z^3+6 a^3 z^3+20 a z^3+14 z^3 a^{-1} -3 a^6 z^2+12 a^2 z^2+11 z^2 a^{-2} +20 z^2-2 a^7 z-6 a^3 z-15 a z-7 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 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 \
8           1-1
6            0
4         11 0
2       11   0
0      2 1   3
-2     231    0
-4    21      1
-6   121      0
-8  22        0
-10 12         1
-12 1          -1
-141           1
Integral Khovanov Homology

(db, data source)

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