L11n285

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_4,15,1,16 X_21,18,22,19 X_9,21,10,20 X_19,5,20,10 X_11,16,12,17 X_17,22,18,11
Gauss code	{1, 4, -3, -6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, -4, -5, 6, 10, -11, 7, -9, 8, -7, 11}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(3)^{2}t(2)^{2}-2t(1)t(3)t(2)^{2}-2t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+2t(2)+2t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$1-q^{-1}+3q^{-2}-3q^{-3}+5q^{-4}-4q^{-5}+5q^{-6}-3q^{-7}+2q^{-8}-q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+z^{4}a^{8}+4z^{2}a^{8}+2a^{8}-z^{6}a^{6}-4z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-z^{6}a^{4}-4z^{4}a^{4}-4z^{2}a^{4}-2a^{4}z^{-2}-4a^{4}+z^{4}a^{2}+4z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{3}-2a^{11}z+2a^{10}z^{4}-4a^{10}z^{2}+2a^{10}+a^{9}z^{7}-4a^{9}z^{5}+8a^{9}z^{3}-4a^{9}z+2a^{8}z^{8}-11a^{8}z^{6}+23a^{8}z^{4}-16a^{8}z^{2}+4a^{8}+a^{7}z^{9}-4a^{7}z^{7}+5a^{7}z^{5}+3a^{6}z^{8}-14a^{6}z^{6}+21a^{6}z^{4}-9a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+a^{5}z^{9}-4a^{5}z^{7}+6a^{5}z^{5}-8a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+a^{4}z^{8}-2a^{4}z^{6}-5a^{4}z^{4}+10a^{4}z^{2}+2a^{4}z^{-2}-7a^{4}+a^{3}z^{7}-3a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+a^{2}z^{6}-5a^{2}z^{4}+7a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

0

-3

3

1

2

-5

1

2

1

0

-7

4

2

-9

1

3

1

-11

3

2

1

-13

1

3

2

-15

1

2

-1

-17

1

-19

1

-1

Integral Khovanov Homology

(db, data source)

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