L10a153

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(2uv^{2}w^{2}-uvw^{2}+2uvw-uw+u+v^{2}w^{3}-v^{2}w^{2}+2vw^{2}-vw+2w\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-3}-2q^{-4}+5q^{-5}-6q^{-6}+9q^{-7}-9q^{-8}+9q^{-9}-7q^{-10}+5q^{-11}-2q^{-12}+q^{-13}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{12}z^{2}+a^{12}z^{-2}+3a^{12}-3a^{10}z^{4}-11a^{10}z^{2}-2a^{10}z^{-2}-10a^{10}+2a^{8}z^{6}+9a^{8}z^{4}+12a^{8}z^{2}+a^{8}z^{-2}+6a^{8}+a^{6}z^{6}+4a^{6}z^{4}+4a^{6}z^{2}+a^{6}$ (db)
Kauffman polynomial	$a^{16}z^{4}-2a^{16}z^{2}+a^{16}+2a^{15}z^{5}-2a^{15}z^{3}+3a^{14}z^{6}-3a^{14}z^{4}+a^{14}z^{2}+3a^{13}z^{7}-2a^{13}z^{5}+3a^{12}z^{8}-6a^{12}z^{6}+11a^{12}z^{4}-11a^{12}z^{2}-a^{12}z^{-2}+6a^{12}+a^{11}z^{9}+4a^{11}z^{7}-17a^{11}z^{5}+24a^{11}z^{3}-13a^{11}z+2a^{11}z^{-1}+6a^{10}z^{8}-21a^{10}z^{6}+34a^{10}z^{4}-32a^{10}z^{2}-2a^{10}z^{-2}+14a^{10}+a^{9}z^{9}+3a^{9}z^{7}-19a^{9}z^{5}+25a^{9}z^{3}-13a^{9}z+2a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+15a^{8}z^{4}-14a^{8}z^{2}-a^{8}z^{-2}+7a^{8}+2a^{7}z^{7}-6a^{7}z^{5}+3a^{7}z^{3}+a^{6}z^{6}-4a^{6}z^{4}+4a^{6}z^{2}-a^{6}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

2

1

-1

-9

3

-11

3

2

-1

-13

6

3

-15

4

0

-17

5

0

-19

3

5

2

-21

2

4

-2

-23

3

-25

1

2

-1

-27

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-7$	$i=-5$
$r=-10$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{2}$
$r=-8$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$