L11a52

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_14,6,15,5 X_10,4,11,3 X_20,12,21,11 X_22,14,5,13 X_12,22,13,21 X_16,9,17,10 X_2,16,3,15 X_8,17,9,18
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -8, 7, -4, 10, -9, 11, -2, 3, -6, 8, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(t(1)-1)(t(2)-1)\left(t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-6q^{11/2}+9q^{9/2}-11q^{7/2}+12q^{5/2}-12q^{3/2}+10{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{7/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+5z^{3}a^{-3}+5za^{-3}+3a^{-3}z^{-1}+z^{7}a^{-1}-az^{5}+4z^{5}a^{-1}-3az^{3}+3z^{3}a^{-1}-3za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}+6z^{5}a^{-7}-4z^{3}a^{-7}+za^{-7}+9z^{6}a^{-6}-13z^{4}a^{-6}+5z^{2}a^{-6}-a^{-6}+10z^{7}a^{-5}-20z^{5}a^{-5}+9z^{3}a^{-5}-2za^{-5}+a^{-5}z^{-1}+8z^{8}a^{-4}-16z^{6}a^{-4}-z^{4}a^{-4}+8z^{2}a^{-4}-3a^{-4}+5z^{9}a^{-3}-9z^{7}a^{-3}-9z^{5}a^{-3}+16z^{3}a^{-3}-9za^{-3}+3a^{-3}z^{-1}+2z^{10}a^{-2}+a^{2}z^{8}-5a^{2}z^{6}-18z^{6}a^{-2}+7a^{2}z^{4}+17z^{4}a^{-2}-2a^{2}z^{2}+2z^{2}a^{-2}-3a^{-2}+3az^{9}+8z^{9}a^{-1}-17az^{7}-36z^{7}a^{-1}+31az^{5}+48z^{5}a^{-1}-18az^{3}-16z^{3}a^{-1}-az-7za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{10}-7z^{8}+2z^{6}+9z^{4}-3z^{2}-2$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

4

1

3

10

5

2

-3

8

6

4

2

6

5

-1

4

6

0

2

6

8

2

0

2

4

-2

2

6

4

-4

1

2

-1

-6

2

-8

1

-1

Integral Khovanov Homology

(db, data source)

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