L11a402

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_20,16,21,15 X_14,8,15,7 X_10,22,5,21 X_18,11,19,12 X_16,9,17,10 X_22,17,11,18 X_8,19,9,20 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 7, -5}, {6, -2, 11, -4, 3, -7, 8, -6, 9, -3, 5, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w-2uvw^{4}+5uvw^{3}-7uvw^{2}+4uvw-uv+uw^{4}-3uw^{3}+4uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-4v^{2}w^{2}+3v^{2}w-v^{2}+vw^{4}-4vw^{3}+7vw^{2}-5vw+2v+w^{3}-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+16q^{2}-20q+25-23q^{-1}+21q^{-2}-15q^{-3}+9q^{-4}-4q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-7a^{2}z^{4}-3z^{4}a^{-2}+10z^{4}+2a^{4}z^{2}-8a^{2}z^{2}-3z^{2}a^{-2}+8z^{2}+a^{4}-2a^{2}+1+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-2a^{6}z^{4}+a^{6}z^{2}+4a^{5}z^{7}-9a^{5}z^{5}+z^{5}a^{-5}+6a^{5}z^{3}-z^{3}a^{-5}-2a^{5}z+7a^{4}z^{8}-15a^{4}z^{6}+4z^{6}a^{-4}+10a^{4}z^{4}-5z^{4}a^{-4}-6a^{4}z^{2}+z^{2}a^{-4}+2a^{4}+6a^{3}z^{9}-3a^{3}z^{7}+8z^{7}a^{-3}-19a^{3}z^{5}-11z^{5}a^{-3}+21a^{3}z^{3}+4z^{3}a^{-3}-7a^{3}z-za^{-3}+2a^{2}z^{10}+17a^{2}z^{8}+11z^{8}a^{-2}-56a^{2}z^{6}-20z^{6}a^{-2}+59a^{2}z^{4}+18z^{4}a^{-2}-28a^{2}z^{2}-9z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+4a^{2}+14az^{9}+8z^{9}a^{-1}-20az^{7}-5z^{7}a^{-1}-9az^{5}-11z^{5}a^{-1}+25az^{3}+15z^{3}a^{-1}-7az-3za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{10}+21z^{8}-64z^{6}+70z^{4}-31z^{2}+2z^{-2}+3$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

10

3

7

3

11

7

-4

1

14

9

5

-1

11

13

2

-3

10

12

-2

-5

6

12

6

-7

3

9

-6

-9

1

6

5

-11

3

-3

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$