L11n144

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_19,4,20,5
Gauss code	{1, -4, -3, 11, 5, -6}, {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, 4, 2, -11, -8, 10, -9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{4}-3t(1)t(2)^{3}+2t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {5}{q^{7/2}}}+q^{5/2}-{\frac {7}{q^{5/2}}}-2q^{3/2}+{\frac {6}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z-az^{5}-3az^{3}+z^{3}a^{-1}-4az-2az^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-3a^{2}z^{8}-z^{8}-3a^{5}z^{7}+2a^{3}z^{7}+5az^{7}-2a^{6}z^{6}+5a^{4}z^{6}+12a^{2}z^{6}+5z^{6}-a^{7}z^{5}+10a^{5}z^{5}-a^{3}z^{5}-14az^{5}-2z^{5}a^{-1}+5a^{6}z^{4}-3a^{4}z^{4}-22a^{2}z^{4}-z^{4}a^{-2}-15z^{4}+3a^{7}z^{3}-11a^{5}z^{3}-4a^{3}z^{3}+14az^{3}+4z^{3}a^{-1}-a^{6}z^{2}-2a^{4}z^{2}+13a^{2}z^{2}+3z^{2}a^{-2}+17z^{2}-a^{7}z+6a^{5}z+2a^{3}z-7az-2za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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

χ

6

1

-1

4

1

2

3

1

-2

0

3

1

2

-2

4

0

-4

3

2

1

-6

2

4

2

-8

2

3

-1

-10

2

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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