L11n220

L11n219

L11n221

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n220 at Knotilus!

Link Presentations

[edit Notes on L11n220's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X15,22,16,9 X2,17,3,18 X21,4,22,5 X14,5,15,6 X20,13,21,14 X11,16,12,17 X6,19,7,20 X18,7,19,8
Gauss code	{1, -5, -3, 6, 7, -10, 11, -2}, {2, -1, -9, 3, 8, -7, -4, 9, 5, -11, 10, -8, -6, 4}

A Braid Representative	{{{braid_table}}}
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)^2 t(2)^5+t(1)^3 t(2)^4-t(1)^2 t(2)^4+t(1)^2 t(2)^3+t(1) t(2)^2-t(1) t(2)+t(2)+t(1)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)-a^{13} z^{-1} +a^{11} z^5+6 a^{11} z^3+9 a^{11} z+3 a^{11} z^{-1} -a^9 z^7-6 a^9 z^5-11 a^9 z^3-8 a^9 z-2 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-10 a^7 z^3-4 a^7 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{15} z+a^{14} z^2+a^{14}+a^{13} z^5-3 a^{13} z^3+3 a^{13} z-a^{13} z^{-1} +a^{12} z^8-6 a^{12} z^6+13 a^{12} z^4-12 a^{12} z^2+3 a^{12}+a^{11} z^9-7 a^{11} z^7+20 a^{11} z^5-29 a^{11} z^3+16 a^{11} z-3 a^{11} z^{-1} +2 a^{10} z^8-11 a^{10} z^6+19 a^{10} z^4-14 a^{10} z^2+3 a^{10}+a^9 z^9-6 a^9 z^7+13 a^9 z^5-16 a^9 z^3+10 a^9 z-2 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+6 a^8 z^4-a^8 z^2+a^7 z^7-6 a^7 z^5+10 a^7 z^3-4 a^7 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -6                 1 1 -8               1 1 0 -10             1     1 -12           1 1     0 -14         2 1       1 -16         1         1 -18     2 2           0 -20                   0 -22 1 2               -1 -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=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\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 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/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.

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