L9n4

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	[edit L9n4 Quick Notes] L9n4 is $9_{43}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9n4's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_11,18,12,5 X_17,10,18,11 X_13,2,14,3
Gauss code	{1, 9, -5, -3}, {-4, -1, 2, 5, -6, 8, -7, 4, -9, -2, 3, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{5}-1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{15/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-2a^{11}z^{-1}+a^{9}z^{5}+6a^{9}z^{3}+10a^{9}z+5a^{9}z^{-1}-a^{7}z^{7}-7a^{7}z^{5}-15a^{7}z^{3}-11a^{7}z-3a^{7}z^{-1}$ (db)
Kauffman polynomial	$-a^{14}-z^{3}a^{11}+4za^{11}-2a^{11}z^{-1}-z^{6}a^{10}+6z^{4}a^{10}-10z^{2}a^{10}+5a^{10}-z^{7}a^{9}+7z^{5}a^{9}-16z^{3}a^{9}+15za^{9}-5a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-10z^{2}a^{8}+5a^{8}-z^{7}a^{7}+7z^{5}a^{7}-15z^{3}a^{7}+11za^{7}-3a^{7}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

-10

1

-12

1

-14

2

1

-16

1

-18

1

0

-20

1

-1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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