L11n5

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,19,12,18 X_13,20,14,21 X_19,5,20,22 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, 11, -2, -3, 7, -9, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{7/2}-3q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}+za^{-3}+az^{5}+4az^{3}-2z^{3}a^{-1}+7az+4az^{-1}-4za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-5a^{2}z^{8}-3z^{8}-a^{5}z^{7}+a^{3}z^{7}-az^{7}-3z^{7}a^{-1}+10a^{4}z^{6}+24a^{2}z^{6}-z^{6}a^{-2}+13z^{6}+5a^{5}z^{5}+13a^{3}z^{5}+21az^{5}+13z^{5}a^{-1}-14a^{4}z^{4}-33a^{2}z^{4}+z^{4}a^{-2}-18z^{4}-7a^{5}z^{3}-26a^{3}z^{3}-37az^{3}-21z^{3}a^{-1}-3z^{3}a^{-3}+7a^{4}z^{2}+17a^{2}z^{2}-z^{2}a^{-4}+11z^{2}+4a^{5}z+16a^{3}z+22az+13za^{-1}+3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

8

1

-1

6

2

4

2

0

2

4

2

0

3

4

1

0

-2

3

4

1

0

-4

2

3

1

-6

1

3

1

-1

-8

1

2

1

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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