L11n130

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_17,3,18,2 X_5,13,6,12 X₆₇₁₈ X_9,16,10,17 X_13,20,14,21 X_15,22,16,7 X_19,4,20,5 X_21,14,22,15
Gauss code	{1, 4, -3, 10, -5, -6}, {6, -1, -7, 3, -2, 5, -8, 11, -9, 7, -4, 2, -10, 8, -11, 9}

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

Polynomial invariants

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

χ

0

1

-1

-2

3

-4

3

2

-1

-6

6

2

4

-8

4

0

-10

5

0

-12

3

4

1

-14

2

5

-3

-16

1

3

2

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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