L6a5

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	[edit L6a5 Quick Notes] L6a5 is $6_{1}^{3}$ in the Rolfsen table of links. It is a closed three-link chain.

[edit L6a5 Further Notes and Views]

Stained glass window of Trinity symbol, Brazil

French coat of arms.

Russian coat of arms.

Russian passport page-number decoration.

Link Presentations

[edit Notes on L6a5's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_12,7,9,8 X_8,11,5,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -5, 2, -6}, {5, -1, 3, -4}, {6, -2, 4, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)t(1)+t(3)t(1)-t(1)-t(2)+t(2)t(3)-t(3)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{-1}-2q^{-2}+3q^{-3}-q^{-4}+3q^{-5}-q^{-6}+q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}-2a^{6}z^{-2}-3a^{6}+2a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{2}$ (db)
Kauffman polynomial	$z^{4}a^{8}-3z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{5}a^{7}-z^{3}a^{7}-3za^{7}+2a^{7}z^{-1}+4z^{4}a^{6}-9z^{2}a^{6}-2a^{6}z^{-2}+5a^{6}+z^{5}a^{5}+z^{3}a^{5}-3za^{5}+2a^{5}z^{-1}+3z^{4}a^{4}-5z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+2z^{3}a^{3}+z^{2}a^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

χ

-1

1

-3

2

1

-1

-5

1

-7

2

-9

3

1

2

-11

1

3

2

-13

0

-15

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$