L11n406

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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_11,22,12,19 X_10,4,11,3 X_5,21,6,20 X_21,5,22,18 X_19,12,20,13 X_14,9,15,10 X_2,14,3,13 X_8,15,9,16
Gauss code	{1, -10, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -11, 9, -5, -4, 8, 10, -9, 11, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$0$ (db)
Jones polynomial	$q^{3}-q^{2}+2q+q^{-1}+2q^{-4}-q^{-5}+q^{-6}-q^{-7}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+4a^{4}+a^{2}z^{-2}-z^{4}-4z^{2}-2z^{-2}-4+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-4a^{7}z+a^{6}z^{8}-6a^{6}z^{6}+11a^{6}z^{4}-10a^{6}z^{2}+4a^{6}+2a^{5}z^{7}-12a^{5}z^{5}+18a^{5}z^{3}-8a^{5}z+2a^{4}z^{8}-14a^{4}z^{6}+30a^{4}z^{4}-28a^{4}z^{2}+8a^{4}+a^{3}z^{9}-6a^{3}z^{7}+8a^{3}z^{5}-2a^{3}z^{3}+2a^{2}z^{8}-13a^{2}z^{6}+z^{6}a^{-2}+22a^{2}z^{4}-5z^{4}a^{-2}-12a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{-2}+az^{9}-6az^{7}+z^{7}a^{-1}+10az^{5}-4z^{5}a^{-1}-10az^{3}+8az+4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+z^{8}-4z^{6}-2z^{4}+12z^{2}+2z^{-2}-7$ (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

1

2

3

4

χ

7

1

5

0

3

2

1

3

1

2

-1

2

5

2

1

-3

1

2

1

-5

1

2

1

0

-7

2

-9

1

-11

1

2

1

0

-13

0

-15

1

-1

Integral Khovanov Homology

(db, data source)

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