L11a11

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-5t(2)^{3}+9t(2)^{2}-5t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {16}{q^{9/2}}}-q^{7/2}+{\frac {23}{q^{7/2}}}+5q^{5/2}-{\frac {27}{q^{5/2}}}-12q^{3/2}+{\frac {27}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+18{\sqrt {q}}-{\frac {25}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+3a^{5}z-3a^{3}z^{5}-6a^{3}z^{3}-4a^{3}z+az^{7}+3az^{5}-z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}+3az+az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+7a^{6}z^{8}-14a^{6}z^{6}+10a^{6}z^{4}-3a^{6}z^{2}+6a^{5}z^{9}-22a^{5}z^{5}+23a^{5}z^{3}-6a^{5}z+2a^{4}z^{10}+20a^{4}z^{8}-53a^{4}z^{6}+41a^{4}z^{4}-10a^{4}z^{2}+15a^{3}z^{9}-9a^{3}z^{7}-34a^{3}z^{5}+z^{5}a^{-3}+35a^{3}z^{3}-8a^{3}z+2a^{2}z^{10}+28a^{2}z^{8}-64a^{2}z^{6}+5z^{6}a^{-2}+41a^{2}z^{4}-3z^{4}a^{-2}-9a^{2}z^{2}+9az^{9}+7az^{7}+12z^{7}a^{-1}-38az^{5}-16z^{5}a^{-1}+26az^{3}+7z^{3}a^{-1}-6az-2za^{-1}+az^{-1}+a^{-1}z^{-1}+15z^{8}-21z^{6}+9z^{4}-3z^{2}-1$ (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

4

-4

4

8

1

7

2

10

4

-6

0

15

8

7

-2

14

12

-2

-4

13

0

-6

10

14

4

-8

6

13

-7

-10

3

10

7

-12

1

6

-5

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{15}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$