L11a58

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+6t(2)^{4}+9t(1)t(2)^{3}-11t(2)^{3}-11t(1)t(2)^{2}+9t(2)^{2}+6t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {17}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-11za^{3}-4a^{3}z^{-1}+z^{7}a+4z^{5}a+8z^{3}a+6za+az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+3a^{8}z^{2}-a^{8}+3a^{7}z^{7}-8a^{7}z^{5}+8a^{7}z^{3}-4a^{7}z+a^{7}z^{-1}+4a^{6}z^{8}-5a^{6}z^{6}-5a^{6}z^{4}+10a^{6}z^{2}-4a^{6}+3a^{5}z^{9}+5a^{5}z^{7}-25a^{5}z^{5}+27a^{5}z^{3}-14a^{5}z+4a^{5}z^{-1}+a^{4}z^{10}+12a^{4}z^{8}-24a^{4}z^{6}+5a^{4}z^{4}+12a^{4}z^{2}-7a^{4}+8a^{3}z^{9}+a^{3}z^{7}-30a^{3}z^{5}+z^{5}a^{-3}+30a^{3}z^{3}-z^{3}a^{-3}-13a^{3}z+4a^{3}z^{-1}+a^{2}z^{10}+17a^{2}z^{8}-33a^{2}z^{6}+4z^{6}a^{-2}+15a^{2}z^{4}-5z^{4}a^{-2}+3a^{2}z^{2}+2z^{2}a^{-2}-4a^{2}+5az^{9}+7az^{7}+8z^{7}a^{-1}-26az^{5}-12z^{5}a^{-1}+19az^{3}+7z^{3}a^{-1}-5az-2za^{-1}+az^{-1}+9z^{8}-11z^{6}+3z^{4}-1$ (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

9

3

-6

0

10

6

4

-2

12

10

-2

-4

10

9

1

-6

7

12

5

-8

6

10

-4

-10

2

8

6

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$