L11n87

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_15,20,16,21 X_9,19,10,18 X_19,9,20,8 X_17,22,18,5 X_21,16,22,17 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -9, 11, -2, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}-uv^{2}-u-v^{3}-v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{7/2}}}-q^{5/2}+{\frac {1}{q^{5/2}}}+2q^{3/2}-{\frac {2}{q^{3/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+3az-2za^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}+5a^{3}z^{7}+4az^{7}-z^{7}a^{-1}+6a^{4}z^{6}+17a^{2}z^{6}+11z^{6}-5a^{3}z^{5}+5z^{5}a^{-1}-9a^{4}z^{4}-25a^{2}z^{4}-16z^{4}+a^{5}z^{3}+a^{3}z^{3}-6az^{3}-6z^{3}a^{-1}+a^{6}z^{2}+4a^{4}z^{2}+10a^{2}z^{2}+7z^{2}+a^{5}z+az+2za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-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

2

1

-8

1

2

-10

1

0

Integral Khovanov Homology

(db, data source)

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