L11n158

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Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_3,10,4,11 X_2,15,3,16 X_14,10,15,9 X_11,19,12,18 X_12,5,13,6 X_6,21,1,22 X_20,14,21,13 X_22,17,7,18 X_19,4,20,5
Gauss code	{1, -4, -3, 11, 7, -8}, {2, -1, 5, 3, -6, -7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v-u^{2}+uv^{4}-uv^{3}+uv^{2}-uv+u-v^{4}+2v^{3}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{3}+2a^{7}z+a^{7}z^{-1}-a^{5}z^{5}-4a^{5}z^{3}-5a^{5}z-a^{5}z^{-1}+a^{3}z^{3}+a^{3}z-az$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-7z^{3}a^{9}+3za^{9}-2z^{8}a^{8}+10z^{6}a^{8}-13z^{4}a^{8}+4z^{2}a^{8}-z^{9}a^{7}+3z^{7}a^{7}+2z^{5}a^{7}-5z^{3}a^{7}-za^{7}+a^{7}z^{-1}-3z^{8}a^{6}+14z^{6}a^{6}-17z^{4}a^{6}+7z^{2}a^{6}-a^{6}-z^{9}a^{5}+4z^{7}a^{5}-4z^{5}a^{5}+4z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-z^{8}a^{4}+4z^{6}a^{4}-4z^{4}a^{4}+2z^{2}a^{4}-z^{5}a^{3}+2z^{3}a^{3}-z^{2}a^{2}-za$ (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

χ

0

1

-2

2

1

-1

-4

1

-6

2

0

-8

2

1

-10

1

2

1

-12

2

0

-14

1

2

1

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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