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

Visit L11n33 at Knotilus!

Link Presentations

[edit Notes on L11n33's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,22 X15,18,16,19 X11,17,12,16 X17,11,18,10 X2,20,3,19
Gauss code {1, -11, 5, -3}, {-6, -1, 2, -5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}
A Braid Representative
A Morse Link Presentation L11n33 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1)}{\sqrt{u} \sqrt{v}} (db)
Jones polynomial 2 q^{9/2}+\frac{1}{q^{9/2}}-3 q^{7/2}-\frac{2}{q^{7/2}}+3 q^{5/2}+\frac{1}{q^{5/2}}-2 q^{3/2}-\frac{1}{q^{3/2}}-q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^5-z^5 a^{-1} -a^3 z^3+5 a z^3-5 z^3 a^{-1} +2 z^3 a^{-3} -2 a^3 z+6 a z-8 z a^{-1} +5 z a^{-3} -z a^{-5} +2 a z^{-1} -4 a^{-1} z^{-1} +3 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial z^7 a^{-5} -5 z^5 a^{-5} +7 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} +a^4 z^6-10 z^6 a^{-4} -4 a^4 z^4+13 z^4 a^{-4} +2 a^4 z^2-7 z^2 a^{-4} + a^{-4} +z^9 a^{-3} +2 a^3 z^7-2 z^7 a^{-3} -10 a^3 z^5-11 z^5 a^{-3} +11 a^3 z^3+24 z^3 a^{-3} -5 a^3 z-16 z a^{-3} +3 a^{-3} z^{-1} +a^2 z^8+4 z^8 a^{-2} -5 a^2 z^6-23 z^6 a^{-2} +3 a^2 z^4+34 z^4 a^{-2} -17 z^2 a^{-2} +a^2+3 a^{-2} +z^9 a^{-1} +3 a z^7-2 z^7 a^{-1} -21 a z^5-17 z^5 a^{-1} +34 a z^3+40 z^3 a^{-1} -17 a z-24 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +3 z^8-19 z^6+28 z^4-12 z^2+2 (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           11
10          1 -1
8         21 1
6       121  0
4      122   -1
2     232    1
0    242     0
-2   123      2
-4  121       0
-6 111        1
-8 1          1
-101           -1
Integral Khovanov Homology

(db, data source)

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