From Knot Atlas
Jump to: navigation, search






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

Visit L11n79 at Knotilus!

Link Presentations

[edit Notes on L11n79's Link Presentations]

Planar diagram presentation X6172 X16,9,17,10 X4,21,1,22 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,22 X15,2,16,3 X20,18,21,17 X18,8,19,7 X8,20,9,19
Gauss code {1, 8, -5, -3}, {-6, -1, 10, -11, 2, 5, -4, 6, -7, 4, -8, -2, 9, -10, 11, -9, 3, 7}
A Braid Representative
A Morse Link Presentation L11n79 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-4 u v^4+6 u v^3-4 u v^2-4 v^3+6 v^2-4 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{4}{q^{7/2}}+6 q^{5/2}-\frac{7}{q^{5/2}}-8 q^{3/2}+\frac{9}{q^{3/2}}+9 \sqrt{q}-\frac{11}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -a^3 z^5-4 a^3 z^3+z^3 a^{-3} -5 a^3 z+2 z a^{-3} -2 a^3 z^{-1} + a^{-3} z^{-1} +a z^7+5 a z^5-2 z^5 a^{-1} +9 a z^3-7 z^3 a^{-1} +7 a z-7 z a^{-1} +3 a z^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-6 a z^7-6 z^7 a^{-1} -3 z^7 a^{-3} -a^4 z^6+6 a^2 z^6+5 z^6 a^{-2} -z^6 a^{-4} +13 z^6+7 a^3 z^5+24 a z^5+26 z^5 a^{-1} +9 z^5 a^{-3} -2 a^4 z^4-10 a^2 z^4+6 z^4 a^{-2} +3 z^4 a^{-4} -5 z^4-3 a^5 z^3-15 a^3 z^3-32 a z^3-27 z^3 a^{-1} -7 z^3 a^{-3} +2 a^4 z^2+7 a^2 z^2-8 z^2 a^{-2} -3 z^2 a^{-4} +4 a^5 z+10 a^3 z+16 a z+13 z a^{-1} +3 z a^{-3} -2 a^2+2 a^{-2} + a^{-4} -a^5 z^{-1} -2 a^3 z^{-1} -3 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} 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 \
10         1-1
8        2 2
6       41 -3
4      42  2
2     54   -1
0    64    2
-2   46     2
-4  35      -2
-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}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.