L11a198

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-4t(2)^{3}+8t(1)^{2}t(2)^{2}-15t(1)t(2)^{2}+8t(2)^{2}-4t(1)^{2}t(2)+8t(1)t(2)-5t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-15q^{5/2}+19q^{3/2}-23{\sqrt {q}}+{\frac {21}{\sqrt {q}}}-{\frac {19}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-4a^{3}z+8az-6za^{-1}+2za^{-3}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-3a^{5}z^{5}+4z^{5}a^{-5}+3a^{5}z^{3}-z^{3}a^{-5}-a^{5}z+4a^{4}z^{8}-14a^{4}z^{6}+9z^{6}a^{-4}+16a^{4}z^{4}-7z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}-a^{4}+5a^{3}z^{9}-12a^{3}z^{7}+14z^{7}a^{-3}+2a^{3}z^{5}-20z^{5}a^{-3}+9a^{3}z^{3}+13z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{3}z^{-1}+2a^{2}z^{10}+10a^{2}z^{8}+14z^{8}a^{-2}-46a^{2}z^{6}-20z^{6}a^{-2}+45a^{2}z^{4}+6z^{4}a^{-2}-9a^{2}z^{2}-3a^{2}+13az^{9}+8z^{9}a^{-1}-24az^{7}+3z^{7}a^{-1}-10az^{5}-39z^{5}a^{-1}+27az^{3}+35z^{3}a^{-1}-11az-12za^{-1}+3az^{-1}+2a^{-1}z^{-1}+2z^{10}+20z^{8}-61z^{6}+43z^{4}-6z^{2}-3$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

6

1

-5

6

9

3

6

4

11

7

-4

2

12

8

4

0

10

12

2

-2

9

11

-2

-4

5

10

5

-6

3

9

-6

-8

1

5

4

-10

3

-3

-12

1

Integral Khovanov Homology

(db, data source)

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