L11a140

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_16,11,17,12 X_12,5,13,6 X_4,18,5,17 X_14,19,15,20 X_20,13,21,14 X_18,22,19,21 X_6,15,1,16
Gauss code	{1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 5, -6, 9, -8, 11, -5, 7, -10, 8, -9, 10, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(t(1)t(2)^{2}-t(2)^{2}-3t(1)t(2)+2t(2)+t(1)-1\right)\left(t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+3t(2)+t(1)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-11q^{3/2}+17{\sqrt {q}}-{\frac {24}{\sqrt {q}}}+{\frac {26}{q^{3/2}}}-{\frac {26}{q^{5/2}}}+{\frac {22}{q^{7/2}}}-{\frac {16}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+3z^{3}a^{5}+3za^{5}-3z^{5}a^{3}-6z^{3}a^{3}-3za^{3}+a^{3}z^{-1}+z^{7}a+3z^{5}a+4z^{3}a-az^{-1}-z^{5}a^{-1}-z^{3}a^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-9a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+7a^{6}z^{8}-14a^{6}z^{6}+9a^{6}z^{4}-2a^{6}z^{2}+6a^{5}z^{9}-a^{5}z^{7}-20a^{5}z^{5}+21a^{5}z^{3}-6a^{5}z+2a^{4}z^{10}+18a^{4}z^{8}-47a^{4}z^{6}+34a^{4}z^{4}-8a^{4}z^{2}+14a^{3}z^{9}-9a^{3}z^{7}-29a^{3}z^{5}+z^{5}a^{-3}+29a^{3}z^{3}-5a^{3}z-a^{3}z^{-1}+2a^{2}z^{10}+24a^{2}z^{8}-53a^{2}z^{6}+5z^{6}a^{-2}+31a^{2}z^{4}-4z^{4}a^{-2}-6a^{2}z^{2}+a^{2}+8az^{9}+7az^{7}+11z^{7}a^{-1}-34az^{5}-15z^{5}a^{-1}+21az^{3}+6z^{3}a^{-1}-az-az^{-1}+13z^{8}-16z^{6}+4z^{4}-z^{2}$ (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

7

1

6

2

10

4

-6

0

14

7

-2

13

11

-2

-4

13

0

-6

10

14

4

-8

6

12

-6

-10

3

10

7

-12

1

6

-5

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$