L11a513

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-u^{2}vw^{3}+2u^{2}vw^{2}-u^{2}vw-u^{2}w^{2}+2u^{2}w-2uv^{2}w^{3}+3uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+4uvw-uv+uw^{2}-3uw+2u-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-2q^{-5}+3q^{4}+5q^{-4}-5q^{3}-7q^{-3}+9q^{2}+11q^{-2}-11q-12q^{-1}+13$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}+4a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-10a^{2}z^{4}-4z^{4}a^{-2}-16a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}-9a^{2}+z^{8}+6z^{6}+13z^{4}+12z^{2}+z^{-2}+5$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+3a^{4}z^{8}+2a^{2}z^{8}+4z^{8}a^{-2}+3z^{8}+2a^{5}z^{7}-2a^{3}z^{7}-14az^{7}-6z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-10a^{4}z^{6}-14a^{2}z^{6}-8z^{6}a^{-2}+3z^{6}a^{-4}-14z^{6}-6a^{5}z^{5}-8a^{3}z^{5}+14az^{5}+7z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}+12a^{4}z^{4}+27a^{2}z^{4}+6z^{4}a^{-2}-7z^{4}a^{-4}+24z^{4}+3a^{5}z^{3}+14a^{3}z^{3}+4az^{3}-2z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4a^{6}z^{2}-11a^{4}z^{2}-25a^{2}z^{2}-z^{2}a^{-2}+3z^{2}a^{-4}-14z^{2}-9a^{3}z-9az-a^{6}+5a^{4}+11a^{2}-a^{-2}+5+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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

11

1

-1

9

2

7

3

1

-2

5

6

2

4

3

6

4

-2

1

7

5

2

-1

7

8

1

-3

4

5

-1

-5

3

7

4

-7

2

4

-2

-9

3

-11

1

2

-1

-13

1

Integral Khovanov Homology

(db, data source)

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