L11n45

L11n44

L11n46

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n45 at Knotilus!

Link Presentations

[edit Notes on L11n45's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X5,14,6,15 X3849 X9,18,10,19 X11,20,12,21 X13,22,14,5 X19,10,20,11 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, 11, -2, -3, 6, -9, 7, -10, 8}

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{(t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right) \left(t(2)^2+t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^9-5 z^3 a^9-7 z a^9-3 a^9 z^{-1} +z^7 a^7+7 z^5 a^7+18 z^3 a^7+20 z a^7+7 a^7 z^{-1} -2 z^5 a^5-10 z^3 a^5-13 z a^5-4 a^5 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^4-3 a^{14} z^2+a^{14}+a^{13} z^5-2 a^{13} z^3+a^{12} z^6-2 a^{12} z^4+a^{12} z^2+a^{11} z^7-3 a^{11} z^5+4 a^{11} z^3+a^{10} z^8-4 a^{10} z^6+6 a^{10} z^4+a^9 z^9-6 a^9 z^7+15 a^9 z^5-16 a^9 z^3+10 a^9 z-3 a^9 z^{-1} +2 a^8 z^8-11 a^8 z^6+24 a^8 z^4-23 a^8 z^2+7 a^8+a^7 z^9-7 a^7 z^7+22 a^7 z^5-36 a^7 z^3+26 a^7 z-7 a^7 z^{-1} +a^6 z^8-6 a^6 z^6+15 a^6 z^4-19 a^6 z^2+7 a^6+3 a^5 z^5-14 a^5 z^3+16 a^5 z-4 a^5 z^{-1} }[/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                   2 2 -6                 1 2 1 -8               2     2 -10             1 1     0 -12           3 2       1 -14         1 1         0 -16       2 3           -1 -18       1             1 -20   1 2               -1 -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}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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.

L11n44 L11n46