L11n345

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_11,19,12,18 X_16,8,17,7 X_17,21,18,20 X_19,5,20,12 X_8,22,9,21 X_22,10,13,9 X_10,14,11,13 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, -3, 6}, {9, -2, 11, -4, -5, 3, -6, 5, 7, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}+t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(1)t(3)t(2)^{2}-t(3)^{2}t(2)+t(2)+t(1)-t(1)t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-q^{8}+q^{7}-q^{6}+3q^{5}-2q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+6z^{4}a^{-8}+11z^{2}a^{-4}-20z^{2}a^{-6}+9z^{2}a^{-8}-z^{2}a^{-10}+7a^{-4}-11a^{-6}+4a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-4z^{2}a^{-12}+a^{-12}-z^{3}a^{-11}-z^{4}a^{-10}-z^{2}a^{-10}+2z^{7}a^{-9}-11z^{5}a^{-9}+11z^{3}a^{-9}+3z^{8}a^{-8}-18z^{6}a^{-8}+29z^{4}a^{-8}-16z^{2}a^{-8}-a^{-8}z^{-2}+5a^{-8}+z^{9}a^{-7}-3z^{7}a^{-7}-7z^{5}a^{-7}+20z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+4z^{8}a^{-6}-25z^{6}a^{-6}+48z^{4}a^{-6}-37z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+z^{9}a^{-5}-5z^{7}a^{-5}+4z^{5}a^{-5}+8z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+17z^{4}a^{-4}-18z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}$ (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

8

χ

21

1

19

1

17

2

1

-1

15

1

0

13

2

3

1

0

11

1

2

9

1

2

1

7

2

1

5

1

3

2

3

0

1

Integral Khovanov Homology

(db, data source)

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