L11a477

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_16,8,5,7 X_18,9,19,10 X_22,15,17,16 X_20,13,21,14 X_12,19,13,20 X_14,21,15,22 X_8,17,9,18 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {9, -4, 7, -6, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -7, 6, -8, 5, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)\left(v^{2}w^{3}-2v^{2}w^{2}+2vw^{2}-2vw+2w-1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+5q^{-8}-8q^{-7}+11q^{-6}-12q^{-5}+13q^{-4}-10q^{-3}+9q^{-2}-q-5q^{-1}+3$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+4a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-2a^{6}z^{6}-10a^{6}z^{4}-17a^{6}z^{2}-2a^{6}z^{-2}-12a^{6}+a^{4}z^{8}+6a^{4}z^{6}+14a^{4}z^{4}+17a^{4}z^{2}+a^{4}z^{-2}+9a^{4}-a^{2}z^{6}-4a^{2}z^{4}-4a^{2}z^{2}-a^{2}$ (db)
Kauffman polynomial	$a^{12}z^{4}-2a^{12}z^{2}+a^{12}+2a^{11}z^{5}-2a^{11}z^{3}+3a^{10}z^{6}-2a^{10}z^{4}+4a^{9}z^{7}-5a^{9}z^{5}+4a^{9}z^{3}+4a^{8}z^{8}-6a^{8}z^{6}+6a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+5a^{8}+3a^{7}z^{9}-3a^{7}z^{7}-5a^{7}z^{5}+12a^{7}z^{3}-10a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+6a^{6}z^{8}-29a^{6}z^{6}+43a^{6}z^{4}-35a^{6}z^{2}-2a^{6}z^{-2}+15a^{6}+6a^{5}z^{9}-17a^{5}z^{7}+7a^{5}z^{5}+11a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+a^{4}z^{10}+5a^{4}z^{8}-33a^{4}z^{6}+51a^{4}z^{4}-36a^{4}z^{2}-a^{4}z^{-2}+12a^{4}+3a^{3}z^{9}-9a^{3}z^{7}+a^{3}z^{5}+9a^{3}z^{3}-3a^{3}z+3a^{2}z^{8}-13a^{2}z^{6}+17a^{2}z^{4}-9a^{2}z^{2}+2a^{2}+az^{7}-4az^{5}+4az^{3}-az$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

3

1

-1

1

2

-1

3

1

-2

-3

6

2

4

-5

6

5

-1

-7

7

4

3

-9

5

6

1

-11

6

7

-1

-13

2

5

3

-15

3

6

-3

-17

1

4

3

-19

1

-1

-21

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$