L11n148

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,19,10,18 X₆₇₁₈ X_19,7,20,22 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_16,12,17,11 X_13,21,14,20 X_21,15,22,14 X_17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, -9, 10, 7, -8, -11, 2, -4, 9, -10, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}+t(2)^{4}-t(1)t(2)^{3}+t(1)t(2)^{2}-t(1)t(2)+t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{5/2}-{\frac {1}{q^{5/2}}}-q^{3/2}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{11/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{5}+a^{5}z^{-1}-z^{5}a-5z^{3}a-7za-2az^{-1}+z^{3}a^{-1}+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{7}+az^{7}+a^{2}z^{6}+z^{6}+7a^{5}z^{5}+a^{3}z^{5}-7az^{5}-z^{5}a^{-1}+a^{4}z^{4}-6a^{2}z^{4}-z^{4}a^{-2}-8z^{4}-14a^{5}z^{3}-4a^{3}z^{3}+13az^{3}+3z^{3}a^{-1}-3a^{4}z^{2}+8a^{2}z^{2}+3z^{2}a^{-2}+14z^{2}+8a^{5}z+2a^{3}z-9az-3za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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

χ

6

1

-1

4

0

2

1

0

1

2

1

-2

1

2

1

-4

1

2

1

0

-6

1

-8

1

0

-10

1

-12

1

Integral Khovanov Homology

(db, data source)

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