L11n404

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_16,9,17,10 X_8,15,9,16 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_2,14,3,13
Gauss code	{1, -11, 7, -5}, {-10, 8, -9, 6}, {-8, -1, 2, -4, 3, -7, -6, 10, 11, -2, 4, -3, 5, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$0$ (db)
Jones polynomial	$q^{2}-q+2+2q^{-2}+q^{-3}-q^{-4}+q^{-5}-q^{-6}+q^{-7}-q^{-8}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-4a^{6}z^{2}-a^{6}z^{-2}-4a^{6}+a^{4}z^{6}+7a^{4}z^{4}+16a^{4}z^{2}+4a^{4}z^{-2}+13a^{4}-a^{2}z^{6}-7a^{2}z^{4}-16a^{2}z^{2}-5a^{2}z^{-2}-14a^{2}+z^{4}+4z^{2}+2z^{-2}+5$ (db)
Kauffman polynomial	$z^{5}a^{9}-4z^{3}a^{9}+2za^{9}+z^{6}a^{8}-4z^{4}a^{8}+2z^{2}a^{8}+z^{7}a^{7}-5z^{5}a^{7}+6z^{3}a^{7}-4za^{7}+a^{7}z^{-1}+z^{8}a^{6}-6z^{6}a^{6}+10z^{4}a^{6}-10z^{2}a^{6}-a^{6}z^{-2}+7a^{6}+3z^{7}a^{5}-21z^{5}a^{5}+40z^{3}a^{5}-25za^{5}+5a^{5}z^{-1}+3z^{8}a^{4}-22z^{6}a^{4}+50z^{4}a^{4}-49z^{2}a^{4}-4a^{4}z^{-2}+22a^{4}+z^{9}a^{3}-4z^{7}a^{3}-7z^{5}a^{3}+34z^{3}a^{3}-31za^{3}+9a^{3}z^{-1}+3z^{8}a^{2}-22z^{6}a^{2}+52z^{4}a^{2}-53z^{2}a^{2}-5a^{2}z^{-2}+23a^{2}+z^{9}a-6z^{7}a+8z^{5}a+4z^{3}a-12za+5az^{-1}+z^{8}-7z^{6}+16z^{4}-16z^{2}-2z^{-2}+9$ (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

5

1

3

0

1

2

1

-1

3

1

2

-3

1

5

2

-5

3

2

3

-7

1

2

1

0

-9

2

4

2

0

-11

1

0

-13

1

2

1

0

-15

0

-17

1

-1

Integral Khovanov Homology

(db, data source)

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