L10n107

L10n106

L10n108

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n107 at Knotilus!
	L10n107 is the "Borromean chain mail" link - it contains two L6a4 configurations without any L2a1 configuration (i.e. no two loops are linked). Compare L10a169.

An indefinitely extended "Borromean chainmail" pattern made up of overlapping L10n107 links; no two circles are directly linked.

"Borromean chain-mail" represented with circles

Represented with minimally-overlapping same-size circles

Link Presentations

[edit Notes on L10n107's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X8493 X2,16,3,15 X16,7,17,8 X9,11,10,14 X13,15,14,20 X19,5,20,10 X11,18,12,19 X4,17,1,18
Gauss code	{1, -4, 3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, -3, -6, 8}, {4, -5, 10, 9, -8, 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{ 0 }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}+\frac{1}{q^{9/2}}-2 q^{7/2}-\frac{2}{q^{7/2}}+q^{5/2}+\frac{1}{q^{5/2}}-2 q^{3/2}-\frac{2}{q^{3/2}}-2 \sqrt{q}-\frac{2}{\sqrt{q}} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^3+z^3 a^{-3} +a^3 z^{-3} - a^{-3} z^{-3} -2 a^3 z+2 z a^{-3} +a z^5-z^5 a^{-1} +5 a z^3-5 z^3 a^{-1} -3 a z^{-3} +3 a^{-1} z^{-3} +6 a z-6 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^8-z^8 a^{-2} -2 z^8-2 a^3 z^7-4 a z^7-4 z^7 a^{-1} -2 z^7 a^{-3} -a^4 z^6+4 a^2 z^6+4 z^6 a^{-2} -z^6 a^{-4} +10 z^6+10 a^3 z^5+26 a z^5+26 z^5 a^{-1} +10 z^5 a^{-3} +4 a^4 z^4+2 a^2 z^4+2 z^4 a^{-2} +4 z^4 a^{-4} -4 z^4-12 a^3 z^3-44 a z^3-44 z^3 a^{-1} -12 z^3 a^{-3} -2 a^4 z^2-8 a^2 z^2-8 z^2 a^{-2} -2 z^2 a^{-4} -12 z^2+8 a^3 z+24 a z+24 z a^{-1} +8 z a^{-3} +1-3 a z^{-1} -3 a^{-1} z^{-1} +3 a^2 z^{-2} +3 a^{-2} z^{-2} +6 z^{-2} -a^3 z^{-3} -3 a z^{-3} -3 a^{-1} z^{-3} - a^{-3} z^{-3} }[/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 5 χ 10                     1 -1 8                   1   1 6               1 1 1   1 4               2 1     1 2           5 2 1       4 0         2 8 2         4 -2       1 2 5           4 -4     1 2               1 -6   1 1 1               1 -8   1                   1 -10 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{ i=2 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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} }[/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 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/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.

L10n106 L10n108