L11a227

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X₄₇₅₈ X_16,5,17,6 X_14,18,15,17 X_6,15,1,16 X_22,14,7,13 X_18,22,19,21 X_2,11,3,12 X_12,3,13,4 X_10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -11, 9, -10, 7, -5, 6, -4, 5, -8, 11, -2, 8, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-3u^{2}v-2uv^{4}+10uv^{3}-15uv^{2}+10uv-2u-3v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	${\frac {23}{q^{9/2}}}-{\frac {24}{q^{7/2}}}+{\frac {21}{q^{5/2}}}+q^{3/2}-{\frac {16}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {14}{q^{13/2}}}-{\frac {20}{q^{11/2}}}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-5a^{7}z-2a^{7}z^{-1}+3a^{5}z^{5}+7a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-3a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{5}+az^{3}-az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-6z^{7}a^{9}+9z^{5}a^{9}-8z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-8z^{8}a^{8}+12z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}-6z^{9}a^{7}-z^{7}a^{7}+19z^{5}a^{7}-26z^{3}a^{7}+12za^{7}-2a^{7}z^{-1}-2z^{10}a^{6}-16z^{8}a^{6}+42z^{6}a^{6}-37z^{4}a^{6}+12z^{2}a^{6}-a^{6}-13z^{9}a^{5}+15z^{7}a^{5}+15z^{5}a^{5}-26z^{3}a^{5}+12za^{5}-2a^{5}z^{-1}-2z^{10}a^{4}-17z^{8}a^{4}+48z^{6}a^{4}-34z^{4}a^{4}+8z^{2}a^{4}-7z^{9}a^{3}+5z^{7}a^{3}+16z^{5}a^{3}-14z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-9z^{8}a^{2}+20z^{6}a^{2}-11z^{4}a^{2}+z^{2}a^{2}-5z^{7}a+10z^{5}a-4z^{3}a-za-z^{6}+z^{4}$ (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

4

0

6

1

-5

-2

10

4

6

-4

12

7

-5

-6

12

9

3

-8

11

12

1

-10

9

12

-3

-12

5

11

6

-14

3

9

-6

-16

1

6

5

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$