L11n328

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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_9,21,10,20 X_19,5,20,12 X_11,13,12,22 X_21,11,22,10 X_4,17,1,18
Gauss code	{1, -4, 3, -11}, {-2, -1, 5, -3, -7, 10, -9, 8}, {-6, 2, 4, -5, 11, 6, -8, 7, -10, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)(vw+1)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{5}-2q^{4}-q^{-4}+4q^{3}+2q^{-3}-4q^{2}-3q^{-2}+6q+5q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-4}+a^{-4}z^{-2}+2a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-3a^{2}z^{2}-7z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-7a^{-2}+z^{6}+5z^{4}+9z^{2}+z^{-2}+7$ (db)
Kauffman polynomial	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+a^{3}z^{7}-2az^{7}-z^{7}a^{-1}+2z^{7}a^{-3}-10a^{2}z^{6}-7z^{6}a^{-2}+z^{6}a^{-4}-18z^{6}-5a^{3}z^{5}-6az^{5}-8z^{5}a^{-1}-7z^{5}a^{-3}+15a^{2}z^{4}+9z^{4}a^{-2}-2z^{4}a^{-4}+26z^{4}+7a^{3}z^{3}+13az^{3}+14z^{3}a^{-1}+10z^{3}a^{-3}+2z^{3}a^{-5}-9a^{2}z^{2}-10z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}a^{-6}-21z^{2}-2a^{3}z-7az-10za^{-1}-6za^{-3}-za^{-5}+3a^{2}+7a^{-2}+a^{-4}-a^{-6}+9+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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

11

1

9

1

-1

7

3

1

2

5

3

0

3

2

1

2

1

3

5

2

-1

2

1

-3

1

3

2

-5

1

2

-1

-7

1

-9

1

-1

Integral Khovanov Homology

(db, data source)

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