L11a432

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Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_18,14,19,13 X_20,9,21,10 X_12,19,5,20 X_22,11,13,12 X_10,21,11,22 X_4,17,1,18
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -3, 7, -10, 9, -8}, {6, -2, 4, -5, 11, -6, 8, -7, 10, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)(t(3)t(2)-t(2)+1)(t(2)t(3)-t(3)+1)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+5q^{-7}-10q^{-6}+16q^{-5}-20q^{-4}+q^{3}+24q^{-3}-4q^{2}-22q^{-2}+8q+20q^{-1}-13$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+2a^{4}z^{6}-5a^{2}z^{6}+z^{6}-a^{6}z^{4}+6a^{4}z^{4}-9a^{2}z^{4}+3z^{4}-a^{6}z^{2}+3a^{4}z^{2}-4a^{2}z^{2}+2z^{2}+a^{6}-4a^{4}+3a^{2}+a^{6}z^{-2}-2a^{4}z^{-2}+a^{2}z^{-2}$ (db)
Kauffman polynomial	$a^{9}z^{5}+5a^{8}z^{6}-5a^{8}z^{4}+a^{8}+10a^{7}z^{7}-14a^{7}z^{5}+3a^{7}z^{3}+11a^{6}z^{8}-14a^{6}z^{6}+3a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-a^{6}+7a^{5}z^{9}+a^{5}z^{7}-16a^{5}z^{5}+5a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+17a^{4}z^{8}-44a^{4}z^{6}+36a^{4}z^{4}-9a^{4}z^{2}+2a^{4}z^{-2}-4a^{4}+13a^{3}z^{9}-22a^{3}z^{7}+7a^{3}z^{5}+3a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+2a^{2}z^{10}+13a^{2}z^{8}-45a^{2}z^{6}+z^{6}a^{-2}+47a^{2}z^{4}-2z^{4}a^{-2}-13a^{2}z^{2}+a^{2}z^{-2}-3a^{2}+6az^{9}-9az^{7}+4z^{7}a^{-1}-2az^{5}-10z^{5}a^{-1}+6az^{3}+5z^{3}a^{-1}+7z^{8}-19z^{6}+17z^{4}-6z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

5

1

4

1

8

3

-5

-1

12

5

7

-3

12

10

-2

-5

12

10

2

-7

8

12

4

-9

8

12

-4

-11

4

10

6

-13

1

6

-5

-15

4

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$