L11n191

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_16,6,17,5 X_15,21,16,20 X_21,19,22,18 X_19,15,20,14 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
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 $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-2t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}-t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+2t(1)t(2)-2t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$5q^{9/2}-6q^{7/2}+7q^{5/2}-{\frac {1}{q^{5/2}}}-7q^{3/2}+{\frac {2}{q^{3/2}}}+q^{13/2}-3q^{11/2}+5{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}-z^{5}a^{-5}-4z^{3}a^{-5}-3za^{-5}+z^{7}a^{-3}+5z^{5}a^{-3}+8z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-8za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-8}+3z^{3}a^{-7}-za^{-7}+z^{6}a^{-6}+z^{4}a^{-6}-z^{2}a^{-6}+3z^{7}a^{-5}-8z^{5}a^{-5}+8z^{3}a^{-5}-3za^{-5}+3z^{8}a^{-4}-8z^{6}a^{-4}+4z^{4}a^{-4}-a^{-4}+z^{9}a^{-3}+3z^{7}a^{-3}-21z^{5}a^{-3}+23z^{3}a^{-3}-8za^{-3}+a^{-3}z^{-1}+5z^{8}a^{-2}-17z^{6}a^{-2}+10z^{4}a^{-2}+4z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+z^{7}a^{-1}-5az^{5}-18z^{5}a^{-1}+9az^{3}+27z^{3}a^{-1}-7az-13za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{8}-8z^{6}+7z^{4}+2z^{2}-3$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

14

1

-1

12

2

10

3

1

-2

8

3

2

1

6

4

3

-1

4

3

0

2

3

5

2

0

2

0

-2

1

4

3

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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