L11a335

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{4}-3u^{3}v^{3}+3u^{3}v^{2}+u^{2}v^{5}-5u^{2}v^{4}+12u^{2}v^{3}-11u^{2}v^{2}+4u^{2}v+4uv^{4}-11uv^{3}+12uv^{2}-5uv+u+3v^{3}-3v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {26}{q^{9/2}}}-{\frac {26}{q^{7/2}}}+{\frac {22}{q^{5/2}}}+q^{3/2}-{\frac {16}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {11}{q^{15/2}}}+{\frac {17}{q^{13/2}}}-{\frac {23}{q^{11/2}}}-4{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+2z^{3}a^{7}+3za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-3z^{5}a^{5}-5z^{3}a^{5}-6za^{5}-3a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-4z^{3}a^{3}-2za^{3}+a^{3}z^{-1}+z^{5}a+2z^{3}a+za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+z^{3}a^{11}-4z^{6}a^{10}+3z^{4}a^{10}-10z^{7}a^{9}+16z^{5}a^{9}-13z^{3}a^{9}+6za^{9}-13z^{8}a^{8}+20z^{6}a^{8}-10z^{4}a^{8}+z^{2}a^{8}-10z^{9}a^{7}+9z^{7}a^{7}+z^{5}a^{7}+3z^{3}a^{7}-6za^{7}+2a^{7}z^{-1}-3z^{10}a^{6}-18z^{8}a^{6}+48z^{6}a^{6}-39z^{4}a^{6}+15z^{2}a^{6}-3a^{6}-17z^{9}a^{5}+31z^{7}a^{5}-20z^{5}a^{5}+16z^{3}a^{5}-12za^{5}+3a^{5}z^{-1}-3z^{10}a^{4}-12z^{8}a^{4}+39z^{6}a^{4}-36z^{4}a^{4}+17z^{2}a^{4}-3a^{4}-7z^{9}a^{3}+8z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+za^{3}+a^{3}z^{-1}-7z^{8}a^{2}+14z^{6}a^{2}-8z^{4}a^{2}+2z^{2}a^{2}-a^{2}-4z^{7}a+9z^{5}a-6z^{3}a+za-z^{6}+2z^{4}-z^{2}$ (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

3

0

6

1

-5

-2

10

3

7

-4

13

7

-6

13

9

4

-8

13

0

-10

10

13

-3

-12

7

13

6

-14

4

10

-6

-16

7

-18

1

4

-3

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}$
$r=-6$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-4$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$