L11n9

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₈₄₉₃ X_5,11,6,10 X_11,20,12,21 X_19,22,20,5 X_13,19,14,18 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -9, 4, 11, -2, 3, 9, -8, 7, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {4}{q^{7/2}}}-q^{5/2}+{\frac {3}{q^{5/2}}}+2q^{3/2}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{11/2}}}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{7}z^{-1}+2a^{5}z+3a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+5az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}-a^{8}+a^{7}z^{3}-a^{7}z+a^{7}z^{-1}+a^{6}z^{4}-2a^{6}+a^{5}z^{7}-4a^{5}z^{5}+6a^{5}z^{3}-6a^{5}z+3a^{5}z^{-1}+2a^{4}z^{8}-10a^{4}z^{6}+15a^{4}z^{4}-8a^{4}z^{2}+a^{3}z^{9}-2a^{3}z^{7}-8a^{3}z^{5}+18a^{3}z^{3}-12a^{3}z+3a^{3}z^{-1}+4a^{2}z^{8}-20a^{2}z^{6}+27a^{2}z^{4}-12a^{2}z^{2}+2a^{2}+az^{9}-2az^{7}+z^{7}a^{-1}-9az^{5}-5z^{5}a^{-1}+20az^{3}+7z^{3}a^{-1}-11az-4za^{-1}+2az^{-1}+a^{-1}z^{-1}+2z^{8}-10z^{6}+13z^{4}-5z^{2}$ (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		\

-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

4

2

1

-4

1

3

1

-6

3

2

1

-8

1

2

3

2

-10

1

2

-1

-12

1

0

-14

1

-1

Integral Khovanov Homology

(db, data source)

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