L11n331

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₈₄₉₃ 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_4,17,1,18
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 $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)(t(2)+t(3))}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-5}+q^{4}+2q^{-4}-q^{3}-3q^{-3}+4q^{2}+5q^{-2}-4q-5q^{-1}+6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}\left(-z^{2}\right)+a^{-4}z^{-2}-a^{4}+a^{-4}+a^{2}z^{4}+2a^{2}z^{2}-2z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-3a^{-2}+z^{4}+z^{2}+z^{-2}+1$ (db)
Kauffman polynomial	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+z^{8}a^{-2}+3z^{8}+2a^{3}z^{7}-3az^{7}-5z^{7}a^{-1}+2a^{4}z^{6}-7a^{2}z^{6}-6z^{6}a^{-2}-15z^{6}+a^{5}z^{5}-5a^{3}z^{5}+2az^{5}+9z^{5}a^{-1}+z^{5}a^{-3}-6a^{4}z^{4}+8a^{2}z^{4}+17z^{4}a^{-2}+z^{4}a^{-4}+30z^{4}-3a^{5}z^{3}+a^{3}z^{3}+5az^{3}-z^{3}a^{-3}+3a^{4}z^{2}-2a^{2}z^{2}-20z^{2}a^{-2}-4z^{2}a^{-4}-21z^{2}+a^{5}z+a^{3}z-3az-6za^{-1}-3za^{-3}-a^{4}+9a^{-2}+4a^{-4}+7+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

0

5

4

1

3

2

0

1

4

2

-1

3

4

1

-3

2

0

-5

1

3

2

-7

1

2

-1

-9

1

-11

1

-1

Integral Khovanov Homology

(db, data source)

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