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

Visit L11n3 at Knotilus!

Link L11n3.
A graph, L11n3.
A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n3's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X5,10,6,11 X8493 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 L11n3 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2-3 v+1\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{7}{q^{5/2}}-q^{3/2}+\frac{6}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+2 \sqrt{q}-\frac{5}{\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^3-5 a^3 z-3 a^3 z^{-1} +2 a z^3+3 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^5 z^9-a^3 z^9-2 a^6 z^8-4 a^4 z^8-2 a^2 z^8-2 a^7 z^7+a^3 z^7-a z^7-a^8 z^6+5 a^6 z^6+14 a^4 z^6+8 a^2 z^6+7 a^7 z^5+9 a^5 z^5+5 a^3 z^5+3 a z^5+4 a^8 z^4+a^6 z^4-16 a^4 z^4-15 a^2 z^4-2 z^4-6 a^7 z^3-13 a^5 z^3-15 a^3 z^3-9 a z^3-z^3 a^{-1} -4 a^8 z^2-4 a^6 z^2+6 a^4 z^2+8 a^2 z^2+2 z^2+3 a^7 z+10 a^5 z+12 a^3 z+7 a z+2 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         11
2        1 -1
0       41 3
-2      43  -1
-4     32   1
-6    34    1
-8   33     0
-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}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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