L11a514

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Planar diagram presentation	X₈₁₉₂ X_16,5,17,6 X_14,3,15,4 X_20,12,21,11 X_18,10,19,9 X_4,15,5,16 X_22,18,13,17 X_12,20,7,19 X_10,22,11,21 X₂₇₃₈ X_6,13,1,14
Gauss code	{1, -10, 3, -6, 2, -11}, {10, -1, 5, -9, 4, -8}, {11, -3, 6, -2, 7, -5, 8, -4, 9, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{2}-u^{2}v^{2}w+u^{2}vw^{3}-3u^{2}vw^{2}+2u^{2}vw-u^{2}w^{3}+2u^{2}w^{2}-2uv^{2}w^{2}+3uv^{2}w-uv^{2}-2uvw^{3}+5uvw^{2}-5uvw+2uv+uw^{3}-3uw^{2}+2uw-2v^{2}w+v^{2}-2vw^{2}+3vw-v+w^{2}-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{4}+3q^{3}-5q^{2}+10q-12+15q^{-1}-15q^{-2}+14q^{-3}-10q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}+a^{6}-2a^{4}z^{4}-3a^{4}z^{2}+a^{4}z^{-2}+a^{4}+a^{2}z^{6}+a^{2}z^{4}-z^{4}a^{-2}-4a^{2}z^{2}-2a^{2}z^{-2}-2z^{2}a^{-2}-6a^{2}+z^{6}+3z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+8a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+7a^{5}z^{7}+6a^{3}z^{7}-11az^{7}-9z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-5a^{4}z^{6}-24a^{2}z^{6}-13z^{6}a^{-2}-26z^{6}+3a^{7}z^{5}-8a^{5}z^{5}-25a^{3}z^{5}-5az^{5}+5z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-8a^{6}z^{4}-a^{4}z^{4}+23a^{2}z^{4}+18z^{4}a^{-2}+33z^{4}-2a^{7}z^{3}+4a^{5}z^{3}+22a^{3}z^{3}+14az^{3}+2z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+7a^{6}z^{2}-a^{4}z^{2}-21a^{2}z^{2}-8z^{2}a^{-2}-20z^{2}-9a^{3}z-9az-2a^{6}+3a^{4}+11a^{2}+7+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

9

1

-1

7

2

5

3

1

-2

3

7

2

5

1

6

4

-2

-1

9

6

3

-3

8

0

-5

6

7

-1

-7

4

8

4

-9

3

6

-3

-11

4

-13

1

3

-2

-15

1

Integral Khovanov Homology

(db, data source)

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