L11n364

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_15,17,16,22 X_9,18,10,19 X_17,8,18,9 X_13,21,14,20 X_21,15,22,14 X_19,5,20,16 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, -9, 7, -8, 4}, {10, -1, 3, 6, -5, -2, 11, -3, -7, 8, -4, 9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)\left(v^{2}w-2vw+2v-1\right)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-5q^{6}+7q^{5}-8q^{4}+9q^{3}-6q^{2}+6q+q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+4z^{4}a^{-4}+6z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-2z^{4}a^{-2}-5z^{2}a^{-2}-2a^{-2}z^{-2}-4a^{-2}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-7z^{4}a^{-8}+2z^{2}a^{-8}+4z^{7}a^{-7}-10z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+3z^{8}a^{-6}-7z^{6}a^{-6}+7z^{4}a^{-6}-5z^{2}a^{-6}+2a^{-6}+z^{9}a^{-5}+z^{7}a^{-5}-6z^{5}a^{-5}+9z^{3}a^{-5}-3za^{-5}+4z^{8}a^{-4}-14z^{6}a^{-4}+27z^{4}a^{-4}-21z^{2}a^{-4}-a^{-4}z^{-2}+7a^{-4}+z^{9}a^{-3}-3z^{7}a^{-3}+7z^{5}a^{-3}-z^{3}a^{-3}-4za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-4z^{6}a^{-2}+14z^{4}a^{-2}-17z^{2}a^{-2}-2a^{-2}z^{-2}+7a^{-2}+2z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}+2a^{-1}z^{-1}+z^{4}-3z^{2}-z^{-2}+3$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

2

13

3

1

-2

11

4

2

9

4

3

-1

7

5

4

1

5

3

6

3

0

1

5

4

-1

1

-1

-3

1

Integral Khovanov Homology

(db, data source)

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