L11n181

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_3,10,4,11 X_5,14,6,15 X_16,8,17,7 X_22,18,7,17 X_15,13,16,12 X_20,10,21,9 X_11,19,12,18 X_13,6,14,1 X_19,4,20,5 X_2,21,3,22
Gauss code	{1, -11, -2, 10, -3, 9}, {4, -1, 7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, -7, 11, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-u^{2}v^{2}+u^{2}v+u^{2}-uv^{3}+3uv^{2}-uv+v^{4}+v^{3}-v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+2q^{5/2}-{\frac {2}{q^{5/2}}}-2q^{3/2}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{11/2}}}+3{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}-a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}-za^{-3}+az^{3}+z^{3}a^{-1}+az+az^{-1}+za^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-7a^{5}z^{5}+14a^{5}z^{3}-9a^{5}z+2a^{5}z^{-1}-2a^{4}z^{4}+6a^{4}z^{2}-3a^{4}-2a^{3}z^{5}+z^{5}a^{-3}+9a^{3}z^{3}-3z^{3}a^{-3}-9a^{3}z+za^{-3}+3a^{3}z^{-1}+2z^{6}a^{-2}-7z^{4}a^{-2}+4a^{2}z^{2}+5z^{2}a^{-2}-3a^{2}+z^{7}a^{-1}+2az^{5}-2z^{5}a^{-1}-3az^{3}-z^{3}a^{-1}+za^{-1}+az^{-1}+2z^{6}-5z^{4}+3z^{2}-1$ (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

χ

8

1

6

1

-1

4

1

0

2

1

-1

0

1

2

1

0

-2

2

3

1

-4

1

2

1

0

-6

2

-8

1

0

-10

1

-12

1

Integral Khovanov Homology

(db, data source)

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