L10n30

L10n29

L10n31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n30 at Knotilus!

Link Presentations

[edit Notes on L10n30's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X15,5,16,20 X7,17,8,16 X17,12,18,13 X9,14,10,15 X13,18,14,19 X19,9,20,8 X2536 X11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -4, 8, -6, 2, -10, 5, -7, 6, -3, 4, -5, 7, -8, 3}

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)^5-3 t(1) t(2)^4+t(2)^4+4 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+4 t(2)^2+t(1) t(2)-3 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{6}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{5/2}}+q^{3/2}-\frac{7}{q^{3/2}}+\frac{2}{q^{13/2}}-\frac{5}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7-a^7 z^{-1} +z^5 a^5+4 z^3 a^5+6 z a^5+4 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-9 z^3 a^3-9 z a^3-4 a^3 z^{-1} +z^5 a+3 z^3 a+2 z a+a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 z^2 a^8+a^8-z^5 a^7-5 z^3 a^7+4 z a^7-a^7 z^{-1} -5 z^6 a^6+11 z^4 a^6-13 z^2 a^6+4 a^6-6 z^7 a^5+18 z^5 a^5-24 z^3 a^5+16 z a^5-4 a^5 z^{-1} -2 z^8 a^4-3 z^6 a^4+21 z^4 a^4-21 z^2 a^4+7 a^4-9 z^7 a^3+29 z^5 a^3-28 z^3 a^3+15 z a^3-4 a^3 z^{-1} -2 z^8 a^2+z^6 a^2+13 z^4 a^2-14 z^2 a^2+4 a^2-3 z^7 a+10 z^5 a-9 z^3 a+3 z a-a z^{-1} -z^6+3 z^4-3 z^2+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 χ 4                 1 -1 2               2   2 0             3 1   -2 -2           4 2     2 -4         4 4       0 -6       4 3         1 -8     2 4           2 -10   3 4             -1 -12   3               3 -14 2                 -2

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}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=3 }[/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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