L11a6

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_12,6,13,5 X₈₄₉₃ X_16,10,17,9 X_20,12,21,11 X_22,18,5,17 X_18,22,19,21 X_10,20,11,19 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -10, 7, -4, 11, -2, 3, -6, 8, -9, 10, -7, 9, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(2v^{4}-4v^{3}+5v^{2}-4v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$22q^{9/2}-22q^{7/2}+18q^{5/2}-15q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-8q^{15/2}+14q^{13/2}-19q^{11/2}+8{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-2z^{3}a^{-7}+a^{-7}z^{-1}+z^{7}a^{-5}+3z^{5}a^{-5}+2z^{3}a^{-5}-2za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+3z^{3}a^{-3}+2za^{-3}-z^{5}a^{-1}-2z^{3}a^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-6z^{4}a^{-10}+2z^{2}a^{-10}+7z^{7}a^{-9}-10z^{5}a^{-9}+4z^{3}a^{-9}+8z^{8}a^{-8}-9z^{6}a^{-8}+5z^{2}a^{-8}-2a^{-8}+6z^{9}a^{-7}-2z^{7}a^{-7}-6z^{5}a^{-7}+4z^{3}a^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}+11z^{8}a^{-6}-22z^{6}a^{-6}+6z^{4}a^{-6}+9z^{2}a^{-6}-5a^{-6}+11z^{9}a^{-5}-15z^{7}a^{-5}+z^{5}a^{-5}+3z^{3}a^{-5}-3za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+9z^{8}a^{-4}-21z^{6}a^{-4}+4z^{4}a^{-4}+7z^{2}a^{-4}-3a^{-4}+5z^{9}a^{-3}-2z^{7}a^{-3}-14z^{5}a^{-3}+12z^{3}a^{-3}-4za^{-3}+6z^{8}a^{-2}-11z^{6}a^{-2}+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-10z^{5}a^{-1}+8z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}+z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

3

-3

16

5

1

4

14

9

3

-6

12

10

5

10

12

9

-3

8

10

0

6

8

12

4

7

10

-3

2

3

10

7

0

1

5

-4

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

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