L11n12

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,10,6,11 X₈₄₉₃ X_13,22,14,5 X_11,20,12,21 X_21,12,22,13 X_19,14,20,15 X_9,18,10,19 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -7, 8, -6, 9, 11, -2, 3, 10, -9, 7, -8, 6}

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

Polynomial invariants

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

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

1

2

1

-6

2

1

3

-8

2

0

-10

3

2

1

-12

2

3

1

0

-14

2

3

-1

-16

1

2

1

0

-18

1

2

-1

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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