L11n226

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_18,11,19,12 X_6,9,7,10 X_7,21,8,20 X_19,1,20,8 X_22,15,9,16 X_3,12,4,13 X_16,6,17,5 X_13,4,14,5 X_14,21,15,22 X_2,18,3,17
Gauss code	{1, -11, -7, 9, 8, -3, -4, 5}, {3, -1, 2, 7, -9, -10, 6, -8, 11, -2, -5, 4, 10, -6}

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

Polynomial invariants

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

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

1

-2

1

3

1

-4

1

-6

1

2

1

0

-8

1

2

1

0

-10

1

0

-12

1

-14

0

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-7$		${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$