L11n26

L11n25

L11n27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n26 at Knotilus!

Link Presentations

[edit Notes on L11n26's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, -11, -2, 3, 8, -10, 9, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{4 t(1) t(2)^3-4 t(1) t(2)^2+t(2)^2+t(1) t(2)-4 t(2)+4}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{5}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} -z^5 a^9-3 z^3 a^9+2 a^9 z^{-1} -2 z^5 a^7-7 z^3 a^7-5 z a^7-2 a^7 z^{-1} -z^5 a^5-3 z^3 a^5-z a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-4 a^{14} z^2+4 a^{14}+a^{13} z^5-a^{13} z^3-a^{13} z-a^{13} z^{-1} +a^{12} z^8-6 a^{12} z^6+18 a^{12} z^4-21 a^{12} z^2+9 a^{12}+a^{11} z^9-5 a^{11} z^7+11 a^{11} z^5-4 a^{11} z^3-3 a^{11} z-a^{11} z^{-1} +3 a^{10} z^8-13 a^{10} z^6+24 a^{10} z^4-14 a^{10} z^2+4 a^{10}+a^9 z^9-2 a^9 z^7-a^9 z^5+11 a^9 z^3-10 a^9 z+2 a^9 z^{-1} +2 a^8 z^8-5 a^8 z^6+2 a^8 z^4+4 a^8 z^2-2 a^8+3 a^7 z^7-10 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +2 a^6 z^6-5 a^6 z^4+a^6 z^2+a^5 z^5-3 a^5 z^3+a^5 z }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 2 1 -1 -8               2     2 -10             3 2     -1 -12           4 2       2 -14         2 4         2 -16       4 3           1 -18       2             2 -20   1 4               -3 -22                     0 -24 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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.

L11n25 L11n27