L11n14

L11n13

L11n15

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n14 at Knotilus!

Link Presentations

[edit Notes on L11n14's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X5,10,6,11 X3849 X11,22,12,5 X13,20,14,21 X19,14,20,15 X21,12,22,13 X9,18,10,19 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, 11, -2, -3, 10, -8, 7, -9, 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{3 (t(1)-1) (t(2)-1)}{\sqrt{t(1)} \sqrt{t(2)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{4}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 (-z)-a^9 z^{-1} +a^7 z^3+2 a^7 z+2 a^7 z^{-1} +a^5 z^3-a^5 z^{-1} +a^3 z^3+a^3 z+a^3 z^{-1} -2 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^8 a^{10}+7 z^6 a^{10}-16 z^4 a^{10}+13 z^2 a^{10}-3 a^{10}-z^9 a^9+6 z^7 a^9-11 z^5 a^9+7 z^3 a^9-3 z a^9+a^9 z^{-1} -3 z^8 a^8+18 z^6 a^8-35 z^4 a^8+26 z^2 a^8-7 a^8-z^9 a^7+4 z^7 a^7-4 z^5 a^7+4 z^3 a^7-6 z a^7+2 a^7 z^{-1} -2 z^8 a^6+9 z^6 a^6-14 z^4 a^6+13 z^2 a^6-4 a^6-2 z^7 a^5+5 z^5 a^5+z^3 a^5-3 z a^5+a^5 z^{-1} -2 z^6 a^4+4 z^4 a^4-2 z^5 a^3+4 z^3 a^3-3 z a^3+a^3 z^{-1} -z^4 a^2-a^2-3 z a+a 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 0                   2 2 -2                 2 2 0 -4               1     1 -6             2 2     0 -8           2 1       1 -10         1 2         1 -12       2 2           0 -14       1             1 -16   1 2               -1 -18                     0 -20 1                   -1

Integral Khovanov Homology (db, data source)

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

L11n13 L11n15