L10n107

See the full Thistlethwaite Link Table (up to 11 crossings).

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.

[edit L10n107 Further Notes and Views]

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 X_5,12,6,13 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_9,11,10,14 X_13,15,14,20 X_19,5,20,10 X_11,18,12,19 X_4,17,1,18
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

6

1

4

2

1

2

5

2

1

4

0

2

8

2

4

-2

1

2

5

4

-4

1

2

1

-6

1

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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