L11a294

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_22,20,9,19 X_18,7,19,8 X_16,5,17,6 X_4,15,5,16 X_6,17,7,18 X_14,22,15,21 X_20,14,21,13 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, -6, 5, -7, 4, -11}, {10, -1, 11, -2, 9, -8, 6, -5, 7, -4, 3, -9, 8, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{5}+t(1)^{3}t(2)^{4}-3t(1)^{2}t(2)^{4}+2t(1)t(2)^{4}-2t(1)^{3}t(2)^{3}+5t(1)^{2}t(2)^{3}-4t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-4t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-2t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)+t(2)+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {11}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-10z^{3}a^{5}-10za^{5}-3a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-4z^{3}a^{3}+a^{3}z^{-1}+z^{5}a+3z^{3}a+za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+6z^{5}a^{9}-4z^{3}a^{9}+2za^{9}-3z^{8}a^{8}+5z^{6}a^{8}-4z^{4}a^{8}+2z^{2}a^{8}-3z^{9}a^{7}+9z^{7}a^{7}-20z^{5}a^{7}+21z^{3}a^{7}-10za^{7}+2a^{7}z^{-1}-z^{10}a^{6}-3z^{8}a^{6}+16z^{6}a^{6}-30z^{4}a^{6}+18z^{2}a^{6}-3a^{6}-6z^{9}a^{5}+22z^{7}a^{5}-39z^{5}a^{5}+34z^{3}a^{5}-16za^{5}+3a^{5}z^{-1}-z^{10}a^{4}-4z^{8}a^{4}+23z^{6}a^{4}-36z^{4}a^{4}+20z^{2}a^{4}-3a^{4}-3z^{9}a^{3}+7z^{7}a^{3}-2z^{5}a^{3}-z^{3}a^{3}-za^{3}+a^{3}z^{-1}-4z^{8}a^{2}+13z^{6}a^{2}-11z^{4}a^{2}+4z^{2}a^{2}-a^{2}-3z^{7}a+10z^{5}a-7z^{3}a+za-z^{6}+3z^{4}-z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

3

1

-2

5

2

3

-4

7

4

-3

-6

6

4

2

-8

6

7

1

-10

5

6

-1

-12

3

6

3

-14

2

5

-3

-16

3

-18

1

2

-1

-20

1

Integral Khovanov Homology

(db, data source)

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