L11n32

L11n31

L11n33

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n32 at Knotilus!

Link Presentations

[edit Notes on L11n32's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X9,14,10,15 X3849 X5,13,6,12 X13,5,14,22 X15,18,16,19 X11,17,12,16 X17,11,18,10 X19,2,20,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, -11, -2, 3, 7}

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) t(2)^5-4 t(1) t(2)^4+5 t(1) t(2)^3-t(2)^3-t(1) t(2)^2+5 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{1}{q^{13/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)+a^5 z^5+4 a^5 z^3+4 a^5 z+2 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-9 a^3 z^3-10 a^3 z-4 a^3 z^{-1} +2 a z^5+8 a z^3-z^3 a^{-1} +8 a z+3 a z^{-1} -3 z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^3 z^9-a z^9-4 a^4 z^8-6 a^2 z^8-2 z^8-5 a^5 z^7-5 a^3 z^7-a z^7-z^7 a^{-1} -2 a^6 z^6+13 a^4 z^6+24 a^2 z^6+9 z^6+18 a^5 z^5+32 a^3 z^5+19 a z^5+5 z^5 a^{-1} +2 a^6 z^4-13 a^4 z^4-27 a^2 z^4-12 z^4-4 a^7 z^3-25 a^5 z^3-44 a^3 z^3-31 a z^3-8 z^3 a^{-1} -a^8 z^2-a^6 z^2+7 a^4 z^2+13 a^2 z^2+6 z^2+2 a^7 z+14 a^5 z+24 a^3 z+17 a z+5 z a^{-1} -a^6-2 a^4-3 a^2-1-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 1   -1 2               3 1   2 0             3 1     -2 -2           4 3       1 -4         4 4         0 -6       3 3           0 -8     2 4             2 -10   2 3               -1 -12   3                 3 -14 1                   -1

Integral Khovanov Homology (db, data source)

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