L11a71

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,17,7 X_18,10,19,9 X_8,18,9,17 X_22,20,5,19 X_20,14,21,13 X_14,22,15,21 X_10,16,11,15 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, 7, -8, 9, -3, 5, -4, 6, -7, 8, -6}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-4uv^{4}+7uv^{3}-8uv^{2}+6uv-3u-3v^{5}+6v^{4}-8v^{3}+7v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$17q^{9/2}-19q^{7/2}+18q^{5/2}-{\frac {1}{q^{5/2}}}-15q^{3/2}+{\frac {2}{q^{3/2}}}+q^{17/2}-4q^{15/2}+9q^{13/2}-13q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}-2z^{5}a^{-5}-4z^{3}a^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-5za^{-1}+2az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-9z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+12z^{7}a^{-7}-16z^{5}a^{-7}+8z^{3}a^{-7}-2za^{-7}+10z^{8}a^{-6}-9z^{6}a^{-6}-5z^{4}a^{-6}+3z^{2}a^{-6}+5z^{9}a^{-5}+7z^{7}a^{-5}-32z^{5}a^{-5}+20z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{10}a^{-4}+16z^{8}a^{-4}-46z^{6}a^{-4}+38z^{4}a^{-4}-15z^{2}a^{-4}+3a^{-4}+8z^{9}a^{-3}-14z^{7}a^{-3}-3z^{5}a^{-3}+5z^{3}a^{-3}+3za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+8z^{8}a^{-2}-34z^{6}a^{-2}+36z^{4}a^{-2}-10z^{2}a^{-2}+3z^{9}a^{-1}+az^{7}-8z^{7}a^{-1}-5az^{5}+4z^{5}a^{-1}+9az^{3}+3z^{3}a^{-1}-7az-4za^{-1}+2az^{-1}+2a^{-1}z^{-1}+2z^{8}-6z^{6}+3z^{4}+4z^{2}-3$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

3

14

6

1

-5

12

7

3

4

10

6

-4

8

9

7

2

6

9

10

1

4

6

9

-3

2

5

10

5

0

2

5

-3

-2

5

-4

1

2

-1

-6

1

Integral Khovanov Homology

(db, data source)

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