L11a293

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{3}-u^{2}v^{2}+u^{2}v+uv^{4}-uv^{3}+uv^{2}-uv+u+v^{3}-v^{2}+v\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{7/2}-3q^{5/2}+5q^{3/2}-9{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {13}{q^{5/2}}}-{\frac {12}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}-3a^{3}z-az^{7}-4az^{5}+z^{5}a^{-1}-4az^{3}+3z^{3}a^{-1}+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}+3a^{8}z^{4}+6a^{7}z^{5}-3a^{7}z^{3}+10a^{6}z^{6}-15a^{6}z^{4}+6a^{6}z^{2}+12a^{5}z^{7}-26a^{5}z^{5}+15a^{5}z^{3}-4a^{5}z+10a^{4}z^{8}-22a^{4}z^{6}+5a^{4}z^{4}+3a^{4}z^{2}+6a^{3}z^{9}-10a^{3}z^{7}-13a^{3}z^{5}+19a^{3}z^{3}-6a^{3}z+2a^{2}z^{10}+4a^{2}z^{8}+z^{8}a^{-2}-34a^{2}z^{6}-5z^{6}a^{-2}+38a^{2}z^{4}+8z^{4}a^{-2}-10a^{2}z^{2}-4z^{2}a^{-2}+9az^{9}+3z^{9}a^{-1}-38az^{7}-16z^{7}a^{-1}+47az^{5}+28z^{5}a^{-1}-17az^{3}-17z^{3}a^{-1}+2za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}-5z^{8}-7z^{6}+23z^{4}-11z^{2}-1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

2

4

3

1

-2

2

6

2

4

0

5

3

-2

9

6

3

-4

6

7

1

-6

6

7

-1

-8

4

6

2

-10

2

6

-4

-12

1

4

3

-14

2

-2

-16

1

Integral Khovanov Homology

(db, data source)

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