L11a148

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_20,11,21,12 X_12,6,13,5 X_4,20,5,19 X_14,18,15,17 X_16,8,17,7 X_22,16,7,15 X_18,14,19,13 X_6,21,1,22
Gauss code	{1, -2, 3, -6, 5, -11}, {8, -1, 2, -3, 4, -5, 10, -7, 9, -8, 7, -10, 6, -4, 11, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-5u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+6uv^{3}-9uv^{2}+6uv-2u-2v^{3}+5v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {3}{q^{9/2}}}-7q^{7/2}+{\frac {6}{q^{7/2}}}+11q^{5/2}-{\frac {11}{q^{5/2}}}-15q^{3/2}+{\frac {14}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+16{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z+za^{-3}+a^{-3}z^{-1}+az^{7}+z^{7}a^{-1}+4az^{5}+3z^{5}a^{-1}+6az^{3}+5az-5za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+z^{7}a^{-5}+3a^{5}z^{5}-3z^{5}a^{-5}-3a^{5}z^{3}+2z^{3}a^{-5}+a^{5}z+4z^{8}a^{-4}+5a^{4}z^{6}-15z^{6}a^{-4}-4a^{4}z^{4}+15z^{4}a^{-4}+a^{4}z^{2}-2z^{2}a^{-4}-a^{-4}+5z^{9}a^{-3}+7a^{3}z^{7}-17z^{7}a^{-3}-9a^{3}z^{5}+15z^{5}a^{-3}+6a^{3}z^{3}-3z^{3}a^{-3}-a^{3}z-za^{-3}+a^{-3}z^{-1}+2z^{10}a^{-2}+7a^{2}z^{8}+3z^{8}a^{-2}-9a^{2}z^{6}-26z^{6}a^{-2}+2a^{2}z^{4}+25z^{4}a^{-2}+2a^{2}z^{2}-2z^{2}a^{-2}-3a^{-2}+5az^{9}+10z^{9}a^{-1}-3az^{7}-28z^{7}a^{-1}-12az^{5}+18z^{5}a^{-1}+15az^{3}+z^{3}a^{-1}-8az-7za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{10}+6z^{8}-25z^{6}+17z^{4}-3$ (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

12

1

10

3

-3

8

4

1

3

6

7

3

-4

4

8

4

2

8

7

-1

0

9

8

1

-2

6

9

3

-4

5

8

-3

-6

2

7

5

-8

1

4

-3

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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