L11n180

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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_9,20,10,21 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	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}-u^{2}v-u^{2}-uv^{2}+uv-u-v^{2}-v+1}{uv}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-q^{5/2}+{\frac {1}{q^{5/2}}}+2q^{3/2}-{\frac {2}{q^{3/2}}}-{\frac {1}{q^{13/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-a^{5}z-a^{5}z^{-1}-a^{3}z^{3}-3a^{3}z+az^{5}+4az^{3}-z^{3}a^{-1}+3az-2za^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-5a^{7}z^{3}+5a^{7}z-a^{7}z^{-1}-2a^{6}z^{2}+a^{6}-3a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+a^{4}z^{8}-6a^{4}z^{6}+8a^{4}z^{4}-3a^{4}z^{2}+a^{3}z^{9}-5a^{3}z^{7}+4a^{3}z^{5}+2a^{3}z^{3}-2a^{3}z+3a^{2}z^{8}-17a^{2}z^{6}+24a^{2}z^{4}-8a^{2}z^{2}+az^{9}-4az^{7}+z^{7}a^{-1}-5z^{5}a^{-1}+6az^{3}+6z^{3}a^{-1}-2az-2za^{-1}+2z^{8}-11z^{6}+16z^{4}-7z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

1

-1

2

1

0

1

2

1

0

-2

1

2

1

0

-4

1

2

1

-6

1

2

1

0

-8

1

-10

1

0

-12

1

-14

1

Integral Khovanov Homology

(db, data source)

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