L11n189

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_5,16,6,17 X_15,21,16,20 X_21,19,22,18 X_19,15,20,14 X₂₇₃₈ X_4,11,5,12 X_17,6,18,1
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, 5, -11, 7, -8, 6, -7, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{3}-2u^{2}v^{2}+u^{2}v+u^{2}-2uv^{3}+5uv^{2}-2uv+v^{4}+v^{3}-2v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+4q^{5/2}-{\frac {3}{q^{5/2}}}-5q^{3/2}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{11/2}}}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{5}+2a^{5}z^{-1}-z^{3}a^{3}-5za^{3}-3a^{3}z^{-1}+2z^{3}a+3za+az^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-3za^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-3z^{8}a^{-2}-4z^{8}-a^{5}z^{7}+4az^{7}-3z^{7}a^{-3}+6a^{2}z^{6}+11z^{6}a^{-2}-z^{6}a^{-4}+18z^{6}+7a^{5}z^{5}+3a^{3}z^{5}-4az^{5}+11z^{5}a^{-1}+11z^{5}a^{-3}+3a^{4}z^{4}-10a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}-25z^{4}-14a^{5}z^{3}-11a^{3}z^{3}-az^{3}-12z^{3}a^{-1}-8z^{3}a^{-3}-7a^{4}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+11z^{2}+10a^{5}z+10a^{3}z+az+2za^{-1}+za^{-3}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

2

1

-1

4

3

2

1

2

3

2

-1

0

1

3

-1

-2

3

4

1

-4

1

2

-1

-6

3

-8

1

0

-10

1

-12

1

Integral Khovanov Homology

(db, data source)

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