L8a21

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	[edit L8a21 Quick Notes] L8a21 is a closed four-link chain. It is $8_{1}^{4}$ in the Rolfsen table of links.

[edit L8a21 Further Notes and Views]

Four linked squares.

Floor decoration in the Biblioteca Medicea Laurenziana, Florence, intended to contain four rings interlinked in this maner (but there is an interlacing error at upper right).

Ornamental circular arcs in square.

Link Presentations

[edit Notes on L8a21's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_16,11,13,12 X_10,3,11,4 X_4,9,1,10 X_14,7,15,8 X_8,13,5,14 X_12,15,9,16
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -8}, {7, -6, 8, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)t(1)-t(2)t(3)t(1)+t(3)t(1)-t(2)t(4)t(1)-t(3)t(4)t(1)+2t(4)t(1)-t(1)-t(2)+2t(2)t(3)-t(3)+t(2)t(4)-t(2)t(3)t(4)+t(3)t(4)-t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	${\frac {4}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {4}{q^{13/2}}}-{\frac {7}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{11}z^{-3}-3a^{9}z^{-3}-4a^{9}z^{-1}+3a^{7}z^{-3}+6a^{7}z+8a^{7}z^{-1}-3a^{5}z^{3}-a^{5}z^{-3}-6a^{5}z-4a^{5}z^{-1}-a^{3}z^{3}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+4z^{3}a^{11}-6za^{11}+4a^{11}z^{-1}-a^{11}z^{-3}-z^{6}a^{10}+6z^{2}a^{10}+3a^{10}z^{-2}-8a^{10}-z^{7}a^{9}-2z^{5}a^{9}+11z^{3}a^{9}-14za^{9}+9a^{9}z^{-1}-3a^{9}z^{-3}-5z^{6}a^{8}+5z^{4}a^{8}+12z^{2}a^{8}+6a^{8}z^{-2}-15a^{8}-z^{7}a^{7}-7z^{5}a^{7}+17z^{3}a^{7}-14za^{7}+9a^{7}z^{-1}-3a^{7}z^{-3}-4z^{6}a^{6}+2z^{4}a^{6}+6z^{2}a^{6}+3a^{6}z^{-2}-8a^{6}-6z^{5}a^{5}+9z^{3}a^{5}-6za^{5}+4a^{5}z^{-1}-a^{5}z^{-3}-3z^{4}a^{4}-z^{3}a^{3}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

3

1

-2

-6

3

-8

1

3

2

-10

6

3

-12

4

7

3

-14

1

-16

4

-18

1

0

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{7}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$