L11a504

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_22,17,13,18 X_20,15,21,16 X_16,21,17,22 X_12,19,7,20 X_18,11,19,12 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 9, -8}, {11, -4, 6, -7, 5, -9, 8, -6, 7, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}+2u^{2}v^{2}w-u^{2}v^{2}-u^{2}vw^{3}+u^{2}vw^{2}-u^{2}vw+u^{2}v-uv^{2}w^{3}+uv^{2}w^{2}-uv^{2}w+uv^{2}+2uvw^{3}-2uvw^{2}+2uvw-2uv-uw^{3}+uw^{2}-uw+u-vw^{3}+vw^{2}-vw+v+w^{3}-2w^{2}+2w-2}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-4}-q^{-5}+5q^{-6}-6q^{-7}+10q^{-8}-11q^{-9}+12q^{-10}-11q^{-11}+9q^{-12}-6q^{-13}+3q^{-14}-q^{-15}$ (db)
Signature	-8 (db)
HOMFLY-PT polynomial	$-a^{14}z^{-2}-a^{14}-a^{12}z^{6}-3a^{12}z^{4}+2a^{12}z^{2}+4a^{12}z^{-2}+9a^{12}+a^{10}z^{8}+4a^{10}z^{6}-a^{10}z^{4}-18a^{10}z^{2}-5a^{10}z^{-2}-19a^{10}+a^{8}z^{8}+7a^{8}z^{6}+18a^{8}z^{4}+21a^{8}z^{2}+2a^{8}z^{-2}+11a^{8}$ (db)
Kauffman polynomial	$z^{3}a^{19}+3z^{4}a^{18}+6z^{5}a^{17}-4z^{3}a^{17}+za^{17}+9z^{6}a^{16}-13z^{4}a^{16}+5z^{2}a^{16}-a^{16}+10z^{7}a^{15}-20z^{5}a^{15}+9z^{3}a^{15}-2za^{15}+a^{15}z^{-1}+8z^{8}a^{14}-17z^{6}a^{14}+3z^{4}a^{14}+3z^{2}a^{14}-a^{14}z^{-2}+4z^{9}a^{13}-3z^{7}a^{13}-23z^{5}a^{13}+33z^{3}a^{13}-19za^{13}+5a^{13}z^{-1}+z^{10}a^{12}+6z^{8}a^{12}-32z^{6}a^{12}+37z^{4}a^{12}-20z^{2}a^{12}-4a^{12}z^{-2}+13a^{12}+5z^{9}a^{11}-16z^{7}a^{11}-z^{5}a^{11}+39z^{3}a^{11}-35za^{11}+9a^{11}z^{-1}+z^{10}a^{10}-z^{8}a^{10}-13z^{6}a^{10}+36z^{4}a^{10}-39z^{2}a^{10}-5a^{10}z^{-2}+22a^{10}+z^{9}a^{9}-3z^{7}a^{9}-4z^{5}a^{9}+20z^{3}a^{9}-19za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-7z^{6}a^{8}+18z^{4}a^{8}-21z^{2}a^{8}-2a^{8}z^{-2}+11a^{8}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-7

1

-9

1

0

-11

4

-13

2

1

-1

-15

8

4

-17

4

3

-1

-19

8

7

1

-21

5

6

1

-23

4

6

-2

-25

2

5

3

-27

1

4

-3

-29

2

-31

1

-1

Integral Khovanov Homology

(db, data source)

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