L9a12

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	[edit L9a12 Quick Notes] L9a12 is $9_{14}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9a12's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,13,5,14 X_14,7,15,8 X_16,9,17,10 X_8,15,9,16 X_10,17,11,18 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -6, 5, -7, 9, -2, 3, -4, 6, -5, 7, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2uv^{4}+2uv^{3}-2uv^{2}+2uv-u-v^{5}+2v^{4}-2v^{3}+2v^{2}-2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {5}{q^{17/2}}}+{\frac {6}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {4}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}(-z)-2a^{11}z^{-1}+3a^{9}z^{3}+9a^{9}z+5a^{9}z^{-1}-2a^{7}z^{5}-8a^{7}z^{3}-9a^{7}z-3a^{7}z^{-1}-a^{5}z^{5}-3a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-a^{14}-2z^{5}a^{13}+3z^{3}a^{13}-2z^{6}a^{12}+z^{4}a^{12}+2z^{2}a^{12}-2z^{7}a^{11}+3z^{5}a^{11}-5z^{3}a^{11}+5za^{11}-2a^{11}z^{-1}-z^{8}a^{10}-z^{6}a^{10}+6z^{4}a^{10}-11z^{2}a^{10}+5a^{10}-5z^{7}a^{9}+17z^{5}a^{9}-26z^{3}a^{9}+16za^{9}-5a^{9}z^{-1}-z^{8}a^{8}-z^{6}a^{8}+9z^{4}a^{8}-12z^{2}a^{8}+5a^{8}-3z^{7}a^{7}+11z^{5}a^{7}-15z^{3}a^{7}+10za^{7}-3a^{7}z^{-1}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

2

1

-1

-8

2

-10

2

0

-12

5

2

3

-14

2

3

1

-16

3

4

-1

-18

1

2

1

-20

1

3

-2

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$