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

Visit L11n36 at Knotilus!

Link Presentations

[edit Notes on L11n36's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X11,14,12,15 X3849 X5,13,6,12 X13,5,14,22 X15,19,16,18 X9,17,10,16 X17,11,18,10 X19,2,20,3
Gauss code {1, 11, -5, -3}, {-6, -1, 2, 5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, -11, -2, 3, 7}
A Braid Representative
A Morse Link Presentation L11n36 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-4 u v^4+5 u v^3-3 u v^2-3 v^3+5 v^2-4 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{9/2}-2 q^{7/2}+4 q^{5/2}-7 q^{3/2}+8 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+5 a z^5-2 z^5 a^{-1} -4 a^3 z^3+9 a z^3-8 z^3 a^{-1} +z^3 a^{-3} +a^5 z-5 a^3 z+8 a z-9 z a^{-1} +3 z a^{-3} +a^5 z^{-1} -2 a^3 z^{-1} +3 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial 3 a^5 z^3-4 a^5 z+a^5 z^{-1} +a^4 z^6+z^6 a^{-4} +2 a^4 z^4-4 z^4 a^{-4} -2 a^4 z^2+4 z^2 a^{-4} - a^{-4} +3 a^3 z^7+2 z^7 a^{-3} -7 a^3 z^5-7 z^5 a^{-3} +13 a^3 z^3+7 z^3 a^{-3} -9 a^3 z-4 z a^{-3} +2 a^3 z^{-1} + a^{-3} z^{-1} +3 a^2 z^8+2 z^8 a^{-2} -8 a^2 z^6-4 z^6 a^{-2} +14 a^2 z^4-3 z^4 a^{-2} -9 a^2 z^2+6 z^2 a^{-2} +2 a^2-2 a^{-2} +a z^9+z^9 a^{-1} +3 a z^7+2 z^7 a^{-1} -15 a z^5-15 z^5 a^{-1} +24 a z^3+21 z^3 a^{-1} -15 a z-14 z a^{-1} +3 a z^{-1} +3 a^{-1} z^{-1} +5 z^8-14 z^6+13 z^4-5 z^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 \
10         1-1
8        1 1
6       31 -2
4      41  3
2     43   -1
0    54    1
-2   45     1
-4  34      -1
-6 14       3
-813        -2
-102         2
Integral Khovanov Homology

(db, data source)

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