L11a430

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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_18,10,19,9 X_4,17,1,18 X_22,19,13,20 X_10,14,11,13 X_12,21,5,22 X_20,11,21,12
Gauss code	{1, -4, 3, -7}, {2, -1, 5, -3, 6, -9, 11, -10}, {9, -2, 4, -5, 7, -6, 8, -11, 10, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)(vw-v+1)(vw-w+1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-19q+24-22q^{-1}+21q^{-2}-15q^{-3}+10q^{-4}-5q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+a^{4}z^{2}-a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-6a^{2}z^{4}-3z^{4}a^{-2}-3a^{2}z^{2}-2z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+3a^{2}+a^{-2}+z^{8}+5z^{6}+9z^{4}+4z^{2}-2z^{-2}-3$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+7a^{3}z^{9}+13az^{9}+6z^{9}a^{-1}+9a^{4}z^{8}+16a^{2}z^{8}+8z^{8}a^{-2}+15z^{8}+5a^{5}z^{7}-7a^{3}z^{7}-19az^{7}+7z^{7}a^{-3}+a^{6}z^{6}-21a^{4}z^{6}-48a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-39z^{6}-10a^{5}z^{5}-11a^{3}z^{5}-2az^{5}-12z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+12a^{4}z^{4}+36a^{2}z^{4}+2z^{4}a^{-2}-6z^{4}a^{-4}+31z^{4}+3a^{5}z^{3}+7a^{3}z^{3}+8az^{3}+9z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-3a^{2}z^{2}+z^{2}a^{-2}+2z^{2}a^{-4}-3z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}-2a^{4}-6a^{2}-2a^{-2}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

5

χ

11

1

-1

9

3

7

5

1

-4

5

9

3

6

3

10

5

-5

1

14

9

5

-1

12

14

2

-3

9

10

-1

-5

6

12

6

-7

4

9

-5

-9

1

6

5

-11

4

-4

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$