L11a354

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X₈₃₉₄ X_16,11,17,12 X_14,8,15,7 X_6,16,7,15 X_22,17,11,18 X_4,20,5,19 X_18,6,19,5 X_20,9,21,10 X_10,21,1,22
Gauss code	{1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {4, -1, 2, -5, 6, -4, 7, -9, 8, -10, 11, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+t(2)t(1)-2t(1)+t(2)^{2}-3t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-8{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {17}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+2z^{5}a^{5}+5z^{3}a^{5}+3za^{5}-z^{7}a^{3}-3z^{5}a^{3}-3z^{3}a^{3}-2za^{3}+2z^{5}a+5z^{3}a+3za+az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+2z^{3}a^{9}-6z^{6}a^{8}+5z^{4}a^{8}-2z^{2}a^{8}-10z^{7}a^{7}+17z^{5}a^{7}-16z^{3}a^{7}+4za^{7}-11z^{8}a^{6}+21z^{6}a^{6}-17z^{4}a^{6}+5z^{2}a^{6}-7z^{9}a^{5}+4z^{7}a^{5}+20z^{5}a^{5}-23z^{3}a^{5}+6za^{5}-2z^{10}a^{4}-13z^{8}a^{4}+46z^{6}a^{4}-39z^{4}a^{4}+12z^{2}a^{4}-11z^{9}a^{3}+24z^{7}a^{3}-2z^{5}a^{3}-11z^{3}a^{3}+4za^{3}-2z^{10}a^{2}-5z^{8}a^{2}+29z^{6}a^{2}-26z^{4}a^{2}+6z^{2}a^{2}-4z^{9}a+9z^{7}a+2z^{5}a-12z^{3}a+6za-az^{-1}-3z^{8}+10z^{6}-10z^{4}+2z^{2}+1-z^{7}a^{-1}+4z^{5}a^{-1}-6z^{3}a^{-1}+4za^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

6

1

5

0

6

2

-4

-2

11

6

5

-4

10

8

-2

-6

9

0

-8

8

10

2

-10

5

9

-4

-12

2

8

6

-14

1

5

-4

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$