L11a159

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+8q^{5/2}-14q^{3/2}+19{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {20}{q^{5/2}}}+{\frac {14}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-5a^{3}z^{3}+z^{3}a^{-3}-6a^{3}z+za^{-3}-2a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+9az^{3}-5z^{3}a^{-1}+8az-5za^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+8a^{5}z^{7}-13a^{5}z^{5}+9a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+9a^{4}z^{8}-12a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-2z^{4}a^{-4}-2a^{4}z^{2}+z^{2}a^{-4}+5a^{3}z^{9}+9a^{3}z^{7}+4z^{7}a^{-3}-36a^{3}z^{5}-10z^{5}a^{-3}+36a^{3}z^{3}+8z^{3}a^{-3}-14a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+20a^{2}z^{8}+6z^{8}a^{-2}-46a^{2}z^{6}-12z^{6}a^{-2}+35a^{2}z^{4}+6z^{4}a^{-2}-10a^{2}z^{2}-z^{2}a^{-2}+a^{2}+9az^{9}+4z^{9}a^{-1}+2az^{7}+5z^{7}a^{-1}-43az^{5}-31z^{5}a^{-1}+48az^{3}+30z^{3}a^{-1}-18az-10za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{10}+17z^{8}-43z^{6}+32z^{4}-9z^{2}$ (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

10

1

-1

8

3

6

5

1

-4

4

9

3

6

2

10

5

-5

0

12

9

3

-2

11

0

-4

9

11

-2

-6

6

12

6

-8

3

8

-5

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

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