L11a7

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(2v^{4}-3v^{3}+3v^{2}-3v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$16q^{9/2}-17q^{7/2}+14q^{5/2}-12q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+3q^{17/2}-5q^{15/2}+10q^{13/2}-14q^{11/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-3z^{3}a^{-7}-za^{-7}+a^{-7}z^{-1}+z^{7}a^{-5}+4z^{5}a^{-5}+5z^{3}a^{-5}+za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+z^{3}a^{-3}-za^{-3}-z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-2z^{3}a^{-11}+3z^{6}a^{-10}-7z^{4}a^{-10}+4z^{2}a^{-10}+4z^{7}a^{-9}-7z^{5}a^{-9}+3z^{3}a^{-9}+4z^{8}a^{-8}-4z^{6}a^{-8}-3z^{4}a^{-8}+7z^{2}a^{-8}-2a^{-8}+4z^{9}a^{-7}-8z^{7}a^{-7}+13z^{5}a^{-7}-9z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}-2z^{6}a^{-6}+z^{4}a^{-6}+6z^{2}a^{-6}-5a^{-6}+9z^{9}a^{-5}-27z^{7}a^{-5}+38z^{5}a^{-5}-24z^{3}a^{-5}+3za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+2z^{8}a^{-4}-12z^{6}a^{-4}+8z^{4}a^{-4}+2z^{2}a^{-4}-3a^{-4}+5z^{9}a^{-3}-11z^{7}a^{-3}+6z^{5}a^{-3}-4z^{3}a^{-3}+2za^{-3}+6z^{8}a^{-2}-16z^{6}a^{-2}+9z^{4}a^{-2}-z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}-11z^{5}a^{-1}+6z^{3}a^{-1}+za^{-1}-a^{-1}z^{-1}+z^{6}-2z^{4}$ (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

2

-2

16

3

1

2

14

7

2

-5

12

7

3

4

10

9

7

-2

8

7

1

6

9

3

4

6

8

-2

2

3

8

5

0

1

4

-3

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

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