L11n6

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₈₄₉₃ 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 $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {1}{q^{9/2}}}-3q^{7/2}+{\frac {2}{q^{7/2}}}+4q^{5/2}-{\frac {4}{q^{5/2}}}-6q^{3/2}+{\frac {6}{q^{3/2}}}+6{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-3za^{3}-3a^{3}z^{-1}+3z^{3}a+6za+4az^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-4za^{-1}-2a^{-1}z^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+z^{6}a^{-4}+2a^{4}z^{4}-3z^{4}a^{-4}-3a^{4}z^{2}+z^{2}a^{-4}+a^{4}+a^{3}z^{7}+3z^{7}a^{-3}-4a^{3}z^{5}-11z^{5}a^{-3}+12a^{3}z^{3}+9z^{3}a^{-3}-12a^{3}z-3za^{-3}+3a^{3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}-9a^{2}z^{6}-10z^{6}a^{-2}+18a^{2}z^{4}+6z^{4}a^{-2}-12a^{2}z^{2}-z^{2}a^{-2}+3a^{2}+a^{-2}+az^{9}+z^{9}a^{-1}+2z^{7}a^{-1}-13az^{5}-20z^{5}a^{-1}+30az^{3}+28z^{3}a^{-1}-20az-13za^{-1}+4az^{-1}+2a^{-1}z^{-1}+5z^{8}-20z^{6}+25z^{4}-11z^{2}+2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

2

1

-1

4

2

1

3

2

0

5

4

1

-2

3

4

1

-4

1

3

-2

-6

1

3

2

-8

1

-1

-10

1

Integral Khovanov Homology

(db, data source)

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