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 $u$, $v$, $w$, ...)	$0$ (db)
Jones polynomial	$q^{9/2}+{\frac {1}{q^{9/2}}}-2q^{7/2}-{\frac {2}{q^{7/2}}}+q^{5/2}+{\frac {1}{q^{5/2}}}-2q^{3/2}-{\frac {2}{q^{3/2}}}-2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{3}z^{3}+z^{3}a^{-3}+a^{3}z^{-3}-a^{-3}z^{-3}-2a^{3}z+2za^{-3}+az^{5}-z^{5}a^{-1}+5az^{3}-5z^{3}a^{-1}-3az^{-3}+3a^{-1}z^{-3}+6az-6za^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{8}-z^{8}a^{-2}-2z^{8}-2a^{3}z^{7}-4az^{7}-4z^{7}a^{-1}-2z^{7}a^{-3}-a^{4}z^{6}+4a^{2}z^{6}+4z^{6}a^{-2}-z^{6}a^{-4}+10z^{6}+10a^{3}z^{5}+26az^{5}+26z^{5}a^{-1}+10z^{5}a^{-3}+4a^{4}z^{4}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}a^{-4}-4z^{4}-12a^{3}z^{3}-44az^{3}-44z^{3}a^{-1}-12z^{3}a^{-3}-2a^{4}z^{2}-8a^{2}z^{2}-8z^{2}a^{-2}-2z^{2}a^{-4}-12z^{2}+8a^{3}z+24az+24za^{-1}+8za^{-3}+1-3az^{-1}-3a^{-1}z^{-1}+3a^{2}z^{-2}+3a^{-2}z^{-2}+6z^{-2}-a^{3}z^{-3}-3az^{-3}-3a^{-1}z^{-3}-a^{-3}z^{-3}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{5}$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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