L10a155

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,14,17,13 X_18,8,19,7 X_10,18,11,17 X_14,9,15,10 X_8,15,9,16 X_16,20,5,19 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {5, -4, 8, -3}, {9, -1, 4, -7, 6, -5, 10, -2, 3, -6, 7, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(uvw^{2}-3uvw+2uv+4uw-2u-2vw^{2}+4vw+2w^{2}-3w+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+10q^{2}-12q+16-16q^{-1}+15q^{-2}-11q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+5z^{2}a^{2}+a^{2}z^{-2}+4a^{2}-z^{6}-3z^{4}-7z^{2}-2z^{-2}-6+z^{4}a^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$2a^{3}z^{9}+2az^{9}+4a^{4}z^{8}+12a^{2}z^{8}+8z^{8}+3a^{5}z^{7}+7a^{3}z^{7}+16az^{7}+12z^{7}a^{-1}+a^{6}z^{6}-7a^{4}z^{6}-21a^{2}z^{6}+10z^{6}a^{-2}-3z^{6}-8a^{5}z^{5}-27a^{3}z^{5}-38az^{5}-15z^{5}a^{-1}+4z^{5}a^{-3}-3a^{6}z^{4}+4a^{2}z^{4}-12z^{4}a^{-2}+z^{4}a^{-4}-12z^{4}+7a^{5}z^{3}+23a^{3}z^{3}+17az^{3}+z^{3}a^{-1}+3a^{6}z^{2}+4a^{4}z^{2}+3a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}-2a^{5}z-6a^{3}z+2az+6za^{-1}-a^{6}-2a^{2}-5a^{-2}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

4

χ

9

1

7

4

1

-3

5

6

3

6

4

-2

1

10

6

4

-1

8

0

-3

7

8

-1

-5

4

8

4

-7

3

7

-4

-9

1

5

4

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }$	${\mathbb {Z} }$