L11a373

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Planar diagram presentation	X_12,1,13,2 X_14,3,15,4 X_18,10,19,9 X_16,6,17,5 X_22,7,11,8 X_6,21,7,22 X_20,15,21,16 X_8,18,9,17 X_4,20,5,19 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -10, 2, -9, 4, -6, 5, -8, 3, -11}, {10, -1, 11, -2, 7, -4, 8, -3, 9, -7, 6, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}v^{3}-u^{4}v^{2}+u^{3}v^{4}-4u^{3}v^{3}+6u^{3}v^{2}-4u^{3}v+u^{3}-2u^{2}v^{4}+7u^{2}v^{3}-11u^{2}v^{2}+7u^{2}v-2u^{2}+uv^{4}-4uv^{3}+6uv^{2}-4uv+u-v^{2}+v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {21}{q^{5/2}}}+{\frac {17}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-3a^{5}z^{3}-3a^{5}z+a^{3}z^{7}+4a^{3}z^{5}+7a^{3}z^{3}+5a^{3}z+a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+2az^{3}-2z^{3}a^{-1}-2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-5a^{5}z^{9}-11a^{3}z^{9}-6az^{9}-5a^{6}z^{8}-7a^{4}z^{8}-11a^{2}z^{8}-9z^{8}-3a^{7}z^{7}+9a^{5}z^{7}+21a^{3}z^{7}+az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+12a^{6}z^{6}+24a^{4}z^{6}+28a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}-5a^{5}z^{5}-16a^{3}z^{5}+11az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-9a^{6}z^{4}-23a^{4}z^{4}-22a^{2}z^{4}+5z^{4}a^{-2}-6z^{4}-6a^{7}z^{3}+2a^{5}z^{3}+9a^{3}z^{3}-7az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-2a^{8}z^{2}+3a^{6}z^{2}+8a^{4}z^{2}+6a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+2a^{7}z-5a^{3}z-az+2za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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

8

1

6

3

-3

4

6

1

5

2

8

3

-5

0

11

6

5

-2

11

9

-2

-4

10

0

-6

8

12

4

-8

5

9

-4

-10

2

8

6

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$