L8a14

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	[edit L8a14 Quick Notes] L8a14 is $8_{1}^{2}$ in the Rolfsen table of links.

[edit L8a14 Further Notes and Views]

Floor of the Stock Exchange [1]	Represented as two interlaced squares	A mosaic seen on an Istanbul floor	Coat of arms of Jaroměř, Czech Republic
Coat of arms of St. Savior, Jersey, Channel Islands, depicting Crown of Thorns religious symbol	Cathedral in Tbilisi, Georgia	Coat of arms of Mozota, Spain	Decorative motif of interlaced squares, San Pancrazio, Florence
Macedonian cross	3D depiction	Arma Christi carving in monastery in Portugal	Handewitt, Schleswig-Holstein oak leaves and acorns
Flower carpet, south India	Flower carpet, south India	Islamic art tile	Moroccan craft
Link on a church of Murato,Corsica	Logo of a spanish hiking trail

Link Presentations

[edit Notes on L8a14's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_14,5,15,6 X_16,7,9,8 X_8,9,1,10 X_4,13,5,14 X_6,15,7,16
Gauss code	{1, -2, 3, -7, 4, -8, 5, -6}, {6, -1, 2, -3, 7, -4, 8, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uv+1)\left(u^{2}v^{2}+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$z^{5}a^{9}+5z^{3}a^{9}+6za^{9}+a^{9}z^{-1}-z^{7}a^{7}-7z^{5}a^{7}-15z^{3}a^{7}-10za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-za^{15}-z^{2}a^{14}-z^{3}a^{13}+za^{13}-z^{4}a^{12}+2z^{2}a^{12}-z^{5}a^{11}+3z^{3}a^{11}-za^{11}-z^{6}a^{10}+4z^{4}a^{10}-3z^{2}a^{10}-z^{7}a^{9}+6z^{5}a^{9}-11z^{3}a^{9}+7za^{9}-a^{9}z^{-1}-z^{6}a^{8}+5z^{4}a^{8}-6z^{2}a^{8}+a^{8}-z^{7}a^{7}+7z^{5}a^{7}-15z^{3}a^{7}+10za^{7}-a^{7}z^{-1}$ (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

-6

1

-8

1

-10

1

-12

0

-14

1

0

-16

0

-18

1

0

-20

0

-22

1

0

-24

1

Integral Khovanov Homology

(db, data source)

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