L11n194

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Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_9,20,10,21 X_21,10,22,11 X_18,16,19,15 X_14,20,15,19 X_2,11,3,12 X_12,3,13,4 X₄₇₅₈ X_22,14,7,13 X_6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, -3, 4, 7, -8, 10, -6, 5, -2, 11, -5, 6, 3, -4, -10}

A Braid Representative

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Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

2

-2

4

3

2

4

2

-2

0

7

3

4

-2

5

0

-4

6

0

-6

4

6

2

-8

2

5

-3

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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