From Knot Atlas
Jump to: navigation, search






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

Visit L11n77 at Knotilus!

Link Presentations

[edit Notes on L11n77's Link Presentations]

Planar diagram presentation X6172 X3,10,4,11 X7,20,8,21 X15,5,16,22 X21,17,22,16 X9,14,10,15 X13,19,14,18 X19,13,20,12 X17,8,18,9 X2536 X11,4,12,1
Gauss code {1, -10, -2, 11}, {10, -1, -3, 9, -6, 2, -11, 8, -7, 6, -4, 5, -9, 7, -8, 3, -5, 4}
A Braid Representative
A Morse Link Presentation L11n77 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^5-4 u v^4+7 u v^3-3 u v^2-3 v^3+7 v^2-4 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{5/2}+3 q^{3/2}-6 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{1}{q^{13/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z a^7+z^5 a^5+4 z^3 a^5+5 z a^5+2 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-11 z a^3-4 a^3 z^{-1} +2 z^5 a+7 z^3 a+7 z a+3 a z^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^2+5 a^7 z^3-a^7 z+3 a^6 z^6-2 a^6 z^4+a^6+8 a^5 z^7-26 a^5 z^5+31 a^5 z^3-14 a^5 z+2 a^5 z^{-1} +7 a^4 z^8-22 a^4 z^6+22 a^4 z^4-12 a^4 z^2+2 a^4+2 a^3 z^9+6 a^3 z^7-42 a^3 z^5+53 a^3 z^3-26 a^3 z+4 a^3 z^{-1} +10 a^2 z^8-37 a^2 z^6+38 a^2 z^4-16 a^2 z^2+3 a^2+2 a z^9-a z^7+z^7 a^{-1} -20 a z^5-4 z^5 a^{-1} +33 a z^3+6 z^3 a^{-1} -17 a z-4 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +3 z^8-12 z^6+14 z^4-5 z^2+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 \
6         11
4        2 -2
2       41 3
0      42  -2
-2     64   2
-4    55    0
-6   45     -1
-8  35      2
-10 24       -2
-12 4        4
-141         -1
Integral Khovanov Homology

(db, data source)

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