L11a164

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_20,16,21,15 X_18,12,19,11 X_12,20,13,19 X_22,18,7,17 X_16,22,17,21 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -7, 11, -4, 5, -9, 8, -6, 7, -5, 9, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+3u^{2}v^{4}-3u^{2}v^{3}+2u^{2}v^{2}+2uv^{5}-4uv^{4}+5uv^{3}-4uv^{2}+2uv+2v^{4}-3v^{3}+3v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+5q^{5/2}-8q^{3/2}+10{\sqrt {q}}-{\frac {13}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-8az^{7}+z^{7}a^{-1}+6a^{3}z^{5}-25az^{5}+6z^{5}a^{-1}+13a^{3}z^{3}-37az^{3}+13z^{3}a^{-1}+11a^{3}z-25az+11za^{-1}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+2a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-6a^{5}z^{5}+4a^{5}z^{3}-2a^{5}z+3a^{4}z^{8}-5a^{4}z^{6}+z^{6}a^{-4}+4a^{4}z^{4}-4z^{4}a^{-4}-2a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-9a^{3}z^{7}+2z^{7}a^{-3}+20a^{3}z^{5}-6z^{5}a^{-3}-22a^{3}z^{3}+3z^{3}a^{-3}+13a^{3}z-3a^{3}z^{-1}+a^{2}z^{10}+3a^{2}z^{8}+3z^{8}a^{-2}-17a^{2}z^{6}-9z^{6}a^{-2}+34a^{2}z^{4}+7z^{4}a^{-2}-23a^{2}z^{2}-3z^{2}a^{-2}+5a^{2}+6az^{9}+3z^{9}a^{-1}-24az^{7}-10z^{7}a^{-1}+49az^{5}+16z^{5}a^{-1}-51az^{3}-19z^{3}a^{-1}+28az+11za^{-1}-5az^{-1}-2a^{-1}z^{-1}+z^{10}+3z^{8}-20z^{6}+37z^{4}-27z^{2}+5$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

4

1

-3

4

1

3

2

6

4

-2

0

7

4

3

-2

6

7

1

-4

5

6

-1

-6

3

6

3

-8

2

5

-3

-10

3

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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