L11n437

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_13,20,14,21 X_9,18,10,19 X_21,10,22,11 X_19,14,20,7 X_5,17,6,16 X_17,22,18,15 X₂₇₃₈ X_4,11,5,12 X_15,1,16,6
Gauss code	{1, -9, 2, -10, -7, 11}, {9, -1, -4, 5, 10, -2, -3, 6}, {-11, 7, -8, 4, -6, 3, -5, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uw+v)\left(uv^{2}w-uv^{2}+u-v^{2}w+w-1\right)}{uv^{3/2}w}}$ (db)
Jones polynomial	$1-q^{-1}+q^{-2}+q^{-4}+q^{-5}+q^{-7}-q^{-8}+q^{-9}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+a^{8}-2a^{6}z^{-2}-3a^{6}-a^{4}z^{6}-5a^{4}z^{4}-5a^{4}z^{2}+a^{4}z^{-2}+a^{2}z^{4}+4a^{2}z^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-5a^{10}z^{4}+5a^{10}z^{2}+a^{9}z^{7}-5a^{9}z^{5}+5a^{9}z^{3}+a^{8}z^{8}-7a^{8}z^{6}+16a^{8}z^{4}-17a^{8}z^{2}-a^{8}z^{-2}+7a^{8}+a^{7}z^{7}-7a^{7}z^{5}+13a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-8a^{6}z^{6}+21a^{6}z^{4}-25a^{6}z^{2}-2a^{6}z^{-2}+11a^{6}+a^{5}z^{7}-7a^{5}z^{5}+13a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}+a^{4}z^{6}-5a^{4}z^{4}+3a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+a^{3}z^{7}-5a^{3}z^{5}+5a^{3}z^{3}+a^{2}z^{6}-5a^{2}z^{4}+6a^{2}z^{2}-2a^{2}$ (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

1

2

χ

1

-1

0

-3

1

2

1

0

-5

1

-7

2

4

1

-9

1

2

-11

1

3

1

-13

1

-15

1

0

-17

1

0

-19

1

Integral Khovanov Homology

(db, data source)

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