L11n330

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_13,18,14,19 X_9,13,10,22 X_11,21,12,20 X_19,5,20,12 X_21,11,22,10 X_17,1,18,4
Gauss code	{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -8, 9}, {-6, 2, 4, -5, -11, 6, -9, 8, -10, 7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-2uv^{2}w+uvw^{3}-2uvw^{2}+uvw-uw^{3}+uw^{2}+v^{3}(-w)+v^{3}-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-vw}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$2q^{4}-2q^{3}+4q^{2}-5q+7-5q^{-1}+5q^{-2}-3q^{-3}+2q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}\left(-z^{2}\right)+a^{-4}z^{-2}-a^{4}+2a^{-4}+a^{2}z^{4}+2a^{2}z^{2}-3z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-5a^{-2}+z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{5}z^{5}-3a^{5}z^{3}+a^{5}z+2a^{4}z^{6}-6a^{4}z^{4}+3z^{4}a^{-4}+3a^{4}z^{2}-10z^{2}a^{-4}-a^{-4}z^{-2}-a^{4}+6a^{-4}+2a^{3}z^{7}+z^{7}a^{-3}-5a^{3}z^{5}-3z^{5}a^{-3}+a^{3}z^{3}+4z^{3}a^{-3}+a^{3}z-5za^{-3}+2a^{-3}z^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}-7a^{2}z^{6}-11z^{6}a^{-2}+9a^{2}z^{4}+27z^{4}a^{-2}-4a^{2}z^{2}-30z^{2}a^{-2}-2a^{-2}z^{-2}+13a^{-2}+az^{9}+z^{9}a^{-1}-2az^{7}-3z^{7}a^{-1}-2az^{5}+z^{5}a^{-1}+9az^{3}+9z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-20z^{6}+39z^{4}-27z^{2}-z^{-2}+9$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

2

7

1

0

5

3

1

2

3

2

1

-1

1

5

3

2

-1

3

5

2

-3

2

0

-5

1

3

2

-7

1

2

-1

-9

1

-11

1

-1

Integral Khovanov Homology

(db, data source)

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