L11n170

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_10,4,11,3 X_2,17,3,18 X_14,5,15,6 X₆₇₁₈ X_16,10,17,9 X_13,21,14,20 X_19,13,20,12 X_22,16,7,15 X_4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 9, -8, -5, 10, -7, 4, 2, -9, 8, 11, -10}

A Braid Representative

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Polynomial invariants

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

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

4

1

-3

10

3

2

1

8

6

4

-2

6

4

3

1

4

3

6

3

2

4

0

4

-2

1

3

-2

-4

1

Integral Khovanov Homology

(db, data source)

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