L11a59

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+6t(2)^{4}+7t(1)t(2)^{3}-7t(2)^{3}-7t(1)t(2)^{2}+7t(2)^{2}+6t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+8q^{7/2}-13q^{5/2}+16q^{3/2}-17{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+2z^{3}a^{-3}-8a^{3}z+za^{-3}-4a^{3}z^{-1}-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+10az^{3}-7z^{3}a^{-1}+10az-6za^{-1}+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-2a^{3}z^{9}-7az^{9}-5z^{9}a^{-1}-2a^{4}z^{8}-5a^{2}z^{8}-10z^{8}a^{-2}-13z^{8}-a^{5}z^{7}+2a^{3}z^{7}+12az^{7}-2z^{7}a^{-1}-11z^{7}a^{-3}+7a^{4}z^{6}+22a^{2}z^{6}+15z^{6}a^{-2}-8z^{6}a^{-4}+38z^{6}+5a^{5}z^{5}+12a^{3}z^{5}+9az^{5}+21z^{5}a^{-1}+15z^{5}a^{-3}-4z^{5}a^{-5}-6a^{4}z^{4}-19a^{2}z^{4}-7z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-28z^{4}-9a^{5}z^{3}-23a^{3}z^{3}-22az^{3}-16z^{3}a^{-1}-6z^{3}a^{-3}+2z^{3}a^{-5}-a^{4}z^{2}+a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+4z^{2}+7a^{5}z+15a^{3}z+12az+4za^{-1}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

5

1

-4

6

8

3

5

4

8

5

-3

2

9

8

1

0

9

0

-2

5

8

-3

-4

4

9

5

-6

2

5

-3

-8

1

5

4

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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