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

Visit L11n2 at Knotilus!

Link L11n2.
A graph, L11n2.
A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n2's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X2,14,3,13
Gauss code {1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -6, 9, 11, -2, -3, 10, -7, 8, -9, 6, -8, 7}
A Braid Representative
A Morse Link Presentation L11n2 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2-4 v+1\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{7}{q^{7/2}}-\frac{9}{q^{5/2}}-2 q^{3/2}+\frac{7}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+4 \sqrt{q}-\frac{6}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+4 a^5 z+3 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-6 a^3 z-3 a^3 z^{-1} +3 a z^3+5 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^5 z^9-a^3 z^9-2 a^6 z^8-5 a^4 z^8-3 a^2 z^8-2 a^7 z^7-a^5 z^7-2 a^3 z^7-3 a z^7-a^8 z^6+5 a^6 z^6+17 a^4 z^6+10 a^2 z^6-z^6+7 a^7 z^5+12 a^5 z^5+14 a^3 z^5+9 a z^5+4 a^8 z^4-20 a^4 z^4-18 a^2 z^4-2 z^4-6 a^7 z^3-16 a^5 z^3-24 a^3 z^3-17 a z^3-3 z^3 a^{-1} -4 a^8 z^2-3 a^6 z^2+8 a^4 z^2+11 a^2 z^2+4 z^2+3 a^7 z+10 a^5 z+14 a^3 z+11 a z+4 z a^{-1} +a^8+2 a^6-2 a^2-a^7 z^{-1} -3 a^5 z^{-1} -3 a^3 z^{-1} -2 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 \
4         22
2        2 -2
0       42 2
-2      54  -1
-4     42   2
-6    35    2
-8   34     -1
-10  13      2
-12 13       -2
-14 1        1
-161         -1
Integral Khovanov Homology

(db, data source)

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