L11a510

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Planar diagram presentation	X₈₁₉₂ X_14,3,15,4 X_22,16,13,15 X_10,20,11,19 X_18,10,19,9 X_20,12,21,11 X_12,22,7,21 X_16,6,17,5 X₂₇₃₈ X_4,13,5,14 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 5, -4, 6, -7}, {10, -2, 3, -8, 11, -5, 4, -6, 7, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-u^{2}vw^{3}+2u^{2}vw^{2}-u^{2}vw-u^{2}w^{2}+2u^{2}w-u^{2}-2uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w+uvw^{3}-2uvw^{2}+2uvw-uv+uw^{2}-2uw+2u+v^{2}w^{3}-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-5q^{7}+8q^{6}-10q^{5}+11q^{4}-10q^{3}+10q^{2}+q^{-2}-6q-2q^{-1}+5$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-10z^{4}a^{-2}+13z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-15z^{2}a^{-2}+13z^{2}a^{-4}-4z^{2}a^{-6}+4z^{2}-8a^{-2}+5a^{-4}-a^{-6}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-11}+3z^{4}a^{-10}-z^{2}a^{-10}+5z^{5}a^{-9}-3z^{3}a^{-9}+7z^{6}a^{-8}-9z^{4}a^{-8}+2z^{2}a^{-8}+8z^{7}a^{-7}-16z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+7z^{8}a^{-6}-17z^{6}a^{-6}+9z^{4}a^{-6}-4z^{2}a^{-6}+a^{-6}+4z^{9}a^{-5}-6z^{7}a^{-5}-13z^{5}a^{-5}+17z^{3}a^{-5}-3za^{-5}+z^{10}a^{-4}+7z^{8}a^{-4}-42z^{6}a^{-4}+59z^{4}a^{-4}-32z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}+6z^{9}a^{-3}-24z^{7}a^{-3}+22z^{5}a^{-3}+4z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+z^{8}a^{-2}-24z^{6}a^{-2}+51z^{4}a^{-2}-38z^{2}a^{-2}-2a^{-2}z^{-2}+12a^{-2}+2z^{9}a^{-1}-10z^{7}a^{-1}+14z^{5}a^{-1}-2z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+z^{8}-6z^{6}+13z^{4}-13z^{2}-z^{-2}+6$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

19

1

-1

17

2

15

3

1

-2

13

5

2

3

11

6

4

-2

9

5

4

1

7

6

7

1

5

4

0

3

7

4

1

2

3

-1

1

4

3

-3

1

-1

-5

1

Integral Khovanov Homology

(db, data source)

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