L11a258

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{4}-2u^{3}v^{3}+2u^{3}v^{2}+u^{2}v^{5}-3u^{2}v^{4}+6u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+2uv^{4}-5uv^{3}+6uv^{2}-3uv+u+2v^{3}-2v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {14}{q^{9/2}}}-{\frac {12}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-10z^{3}a^{5}-9za^{5}-3a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-5z^{3}a^{3}-2za^{3}+a^{3}z^{-1}+z^{5}a+3z^{3}a+2za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+6z^{5}a^{9}-5z^{3}a^{9}+3za^{9}-3z^{8}a^{8}+4z^{6}a^{8}-2z^{4}a^{8}-3z^{9}a^{7}+7z^{7}a^{7}-14z^{5}a^{7}+13z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-z^{10}a^{6}-4z^{8}a^{6}+15z^{6}a^{6}-23z^{4}a^{6}+12z^{2}a^{6}-3a^{6}-6z^{9}a^{5}+17z^{7}a^{5}-26z^{5}a^{5}+24z^{3}a^{5}-14za^{5}+3a^{5}z^{-1}-z^{10}a^{4}-5z^{8}a^{4}+20z^{6}a^{4}-25z^{4}a^{4}+15z^{2}a^{4}-3a^{4}-3z^{9}a^{3}+4z^{7}a^{3}+4z^{5}a^{3}-4z^{3}a^{3}+za^{3}+a^{3}z^{-1}-4z^{8}a^{2}+10z^{6}a^{2}-5z^{4}a^{2}+2z^{2}a^{2}-a^{2}-3z^{7}a+9z^{5}a-7z^{3}a+2za-z^{6}+3z^{4}-2z^{2}$ (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

4

1

-3

-2

6

2

4

-4

7

5

-2

-6

7

5

2

-8

7

0

-10

5

7

-2

-12

3

7

4

-14

2

5

-3

-16

3

-18

1

2

-1

-20

1

Integral Khovanov Homology

(db, data source)

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