L11n160

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Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_10,4,11,3 X_2,15,3,16 X_14,10,15,9 X_11,19,12,18 X_5,13,6,12 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 $u$ , $v$ , $w$ , ...)	$-{\frac {(uv-u-v+2)(2uv-u-v+1)}{uv}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+5q^{5/2}-7q^{3/2}+8{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{3}+z^{3}a^{-3}+a^{3}z+za^{-3}+a^{3}z^{-1}-az^{5}-z^{5}a^{-1}-2az^{3}-2z^{3}a^{-1}-2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-3a^{2}z^{8}-4z^{8}a^{-2}-7z^{8}-a^{3}z^{7}+5az^{7}+3z^{7}a^{-1}-3z^{7}a^{-3}+11a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+26z^{6}-3az^{5}+7z^{5}a^{-1}+10z^{5}a^{-3}-3a^{4}z^{4}-18a^{2}z^{4}-12z^{4}a^{-2}+3z^{4}a^{-4}-30z^{4}-a^{5}z^{3}-a^{3}z^{3}-3az^{3}-10z^{3}a^{-1}-7z^{3}a^{-3}+3a^{4}z^{2}+8a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}+10z^{2}+a^{5}z+3za^{-1}+2za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

3

1

-2

4

2

4

3

-1

0

5

4

1

-2

3

5

2

-4

3

4

-1

-6

1

4

3

-8

2

-2

-10

1

Integral Khovanov Homology

(db, data source)

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