L11a47

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_18,9,19,10 X_8,17,9,18 X_4,19,1,20 X_12,6,13,5 X_10,4,11,3 X_22,14,5,13 X_14,22,15,21 X_20,12,21,11 X_2,16,3,15
Gauss code	{1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 10, -6, 8, -9, 11, -2, 4, -3, 5, -10, 9, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {4}{q^{9/2}}}-6q^{7/2}+{\frac {8}{q^{7/2}}}+10q^{5/2}-{\frac {13}{q^{5/2}}}-15q^{3/2}+{\frac {16}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-2a^{3}z^{3}-3z^{3}a^{-3}-2za^{-3}+a^{3}z^{-1}-a^{-3}z^{-1}+az^{7}+z^{7}a^{-1}+3az^{5}+4z^{5}a^{-1}+az^{3}+6z^{3}a^{-1}-3az+5za^{-1}-2az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+z^{7}a^{-5}+4a^{5}z^{5}-4z^{5}a^{-5}-2a^{5}z^{3}+4z^{3}a^{-5}+3z^{8}a^{-4}+8a^{4}z^{6}-12z^{6}a^{-4}-7a^{4}z^{4}+14z^{4}a^{-4}+a^{4}z^{2}-4z^{2}a^{-4}+4z^{9}a^{-3}+11a^{3}z^{7}-15z^{7}a^{-3}-15a^{3}z^{5}+20z^{5}a^{-3}+6a^{3}z^{3}-14z^{3}a^{-3}+a^{3}z+5za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2z^{10}a^{-2}+10a^{2}z^{8}-14a^{2}z^{6}-14z^{6}a^{-2}+4a^{2}z^{4}+16z^{4}a^{-2}+a^{2}z^{2}-5z^{2}a^{-2}+6az^{9}+10z^{9}a^{-1}-4az^{7}-31z^{7}a^{-1}-6az^{5}+37z^{5}a^{-1}-3az^{3}-29z^{3}a^{-1}+8az+12za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+7z^{8}-24z^{6}+14z^{4}-z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

2

-2

8

4

1

3

6

2

-4

4

9

4

5

2

8

6

-2

0

10

9

1

-2

8

10

2

-4

5

8

-3

-6

3

8

5

-8

1

5

-4

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

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