L11a366

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

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Polynomial invariants

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

12

1

-1

10

2

8

4

1

-3

6

5

2

3

4

6

4

-2

2

7

5

2

0

5

7

2

-2

5

6

-1

-4

3

6

3

-6

1

4

-3

-8

1

3

2

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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