L11a101

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)\left(2v^{2}-v+2\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+13{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {16}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {6}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+6a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-2a^{5}z+3a^{3}z-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}+4a^{6}z^{8}-9a^{6}z^{6}+5a^{6}z^{4}-a^{6}z^{2}+4a^{5}z^{9}-8a^{5}z^{7}+9a^{5}z^{5}-10a^{5}z^{3}+4a^{5}z+2a^{4}z^{10}+2a^{4}z^{8}-8a^{4}z^{6}+7a^{4}z^{4}-4a^{4}z^{2}+10a^{3}z^{9}-25a^{3}z^{7}+34a^{3}z^{5}+z^{5}a^{-3}-25a^{3}z^{3}-z^{3}a^{-3}+6a^{3}z+2a^{2}z^{10}+7a^{2}z^{8}-18a^{2}z^{6}+4z^{6}a^{-2}+14a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+6az^{9}-6az^{7}+8z^{7}a^{-1}+az^{5}-14z^{5}a^{-1}+7z^{3}a^{-1}-2za^{-1}+az^{-1}+a^{-1}z^{-1}+9z^{8}-16z^{6}+10z^{4}-2z^{2}-1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

7

3

-4

0

11

6

5

-2

10

9

-1

-4

9

0

-6

7

10

3

-8

4

9

-5

-10

2

7

5

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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