L11n450

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_11,20,12,21 X_13,19,14,22 X_21,18,22,9 X_17,13,18,12 X_15,8,16,5 X_7,14,8,15 X_19,17,20,16 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -8, 7}, {-9, 3, -5, 4}, {-11, 2, -3, 6, -4, 8, -7, 9, -6, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)(x-1)^{2}(uvx+u(-v)+u-vx+x-1)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-13q^{3/2}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {4}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}-3a^{3}z^{3}+8az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-7a^{3}z+12az-8za^{-1}+2za^{-3}+2a^{5}z^{-1}-8a^{3}z^{-1}+11az^{-1}-6a^{-1}z^{-1}+a^{-3}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$10a^{5}z^{3}-a^{5}z^{-3}-11a^{5}z+5a^{5}z^{-1}+6a^{4}z^{6}+z^{6}a^{-4}-4a^{4}z^{4}-3z^{4}a^{-4}+9a^{4}z^{2}+3z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+13a^{3}z^{7}+3z^{7}a^{-3}-31a^{3}z^{5}-8z^{5}a^{-3}+44a^{3}z^{3}+9z^{3}a^{-3}-3a^{3}z^{-3}-34a^{3}z-6za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+9a^{2}z^{8}+4z^{8}a^{-2}-9a^{2}z^{6}-5z^{6}a^{-2}-9a^{2}z^{4}-7z^{4}a^{-2}+24a^{2}z^{2}+14z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+2az^{9}+2z^{9}a^{-1}+20az^{7}+10z^{7}a^{-1}-62az^{5}-39z^{5}a^{-1}+67az^{3}+42z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-44az-27za^{-1}+18az^{-1}+11a^{-1}z^{-1}+13z^{8}-21z^{6}-9z^{4}+26z^{2}+3z^{-2}-18$ (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

5

1

-4

4

8

2

6

2

6

5

-1

0

12

8

4

-2

8

12

4

-4

6

0

-6

2

8

6

-8

2

6

-4

-10

4

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-4$	${\mathbb {Z} }^{4}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$