L11a390

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,7,17,8 X_8,15,5,16 X_18,12,19,11 X_20,13,21,14 X_22,20,9,19 X_12,21,13,22 X_14,18,15,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -8, 6, -9, 4, -3, 9, -5, 7, -6, 8, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{3}-4uvw^{2}+6uvw-2uv-2uw^{3}+7uw^{2}-7uw+2u-2vw^{3}+7vw^{2}-7vw+2v+2w^{3}-6w^{2}+4w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+6q^{-7}-10q^{-6}+16q^{-5}-19q^{-4}+21q^{-3}-q^{2}-18q^{-2}+5q+15q^{-1}-10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-3a^{8}z^{-2}-4a^{8}+6a^{6}z^{2}+4a^{6}z^{-2}+8a^{6}-4a^{4}z^{4}-7a^{4}z^{2}-3a^{4}z^{-2}-7a^{4}+a^{2}z^{6}+2a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+3a^{2}-z^{4}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+6a^{10}z^{2}+a^{10}z^{-2}-4a^{10}+2a^{9}z^{7}-5a^{9}z^{5}+3a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+2a^{8}z^{8}+2a^{8}z^{6}-18a^{8}z^{4}+25a^{8}z^{2}+3a^{8}z^{-2}-14a^{8}+2a^{7}z^{9}+3a^{7}z^{7}-11a^{7}z^{5}+6a^{7}z^{3}+a^{7}z-a^{7}z^{-1}+a^{6}z^{10}+6a^{6}z^{8}-5a^{6}z^{6}-19a^{6}z^{4}+35a^{6}z^{2}+4a^{6}z^{-2}-21a^{6}+7a^{5}z^{9}-3a^{5}z^{7}-12a^{5}z^{5}+10a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{4}z^{10}+14a^{4}z^{8}-23a^{4}z^{6}-3a^{4}z^{4}+23a^{4}z^{2}+3a^{4}z^{-2}-14a^{4}+5a^{3}z^{9}+6a^{3}z^{7}-22a^{3}z^{5}+10a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+10a^{2}z^{8}-12a^{2}z^{6}-3a^{2}z^{4}+7a^{2}z^{2}+a^{2}z^{-2}-4a^{2}+10az^{7}-15az^{5}+z^{5}a^{-1}+3az^{3}+5z^{6}-5z^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

6

1

-5

-1

9

4

5

-3

10

7

-3

-5

11

8

3

-7

8

10

2

-9

8

11

-3

-11

5

11

6

-13

1

5

-4

-15

1

5

4

-17

1

-1

-19

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$