L11n142

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, -7, 3, 2, -5, 9, -10, 8, 7, -4, -2, 11, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{2}-2u^{2}v-3uv^{2}+7uv-3u-2v+2}{uv}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+4{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-3z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}-z^{5}a-2z^{3}a+2az^{-1}+z^{3}a^{-1}+za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-9a^{5}z^{5}+8a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+16a^{4}z^{4}-8a^{4}z^{2}+a^{3}z^{9}-8a^{3}z^{5}+18a^{3}z^{3}-10a^{3}z+2a^{3}z^{-1}+4a^{2}z^{8}-16a^{2}z^{6}+27a^{2}z^{4}+z^{4}a^{-2}-13a^{2}z^{2}-2z^{2}a^{-2}+a^{2}+az^{9}-3az^{7}+5az^{5}+3z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}-4az+za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{8}-3z^{6}+7z^{4}-6z^{2}$ (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

χ

6

1

-1

4

2

1

-1

0

5

2

3

-2

3

0

-4

4

0

-6

3

4

1

-8

1

3

-2

-10

1

3

2

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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