L11n205

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_7,16,8,17 X_18,12,19,11 X_2,19,3,20 X_3,12,4,13 X_13,21,14,20 X_14,5,15,6 X_6,9,7,10 X_22,16,9,15 X_17,8,18,1 X_21,4,22,5
Gauss code	{1, -4, -5, 11, 7, -8, -2, 10}, {8, -1, 3, 5, -6, -7, 9, 2, -10, -3, 4, 6, -11, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(uv^{2}+1\right)\left(u^{2}+v^{3}\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+4za^{7}+2a^{7}z^{-1}-z^{5}a^{5}-6z^{3}a^{5}-9za^{5}-3a^{5}z^{-1}+za^{3}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-7a^{9}z-a^{7}z^{3}+4a^{7}z-2a^{7}z^{-1}-a^{6}z^{6}+6a^{6}z^{4}-9a^{6}z^{2}+3a^{6}-a^{5}z^{7}+7a^{5}z^{5}-15a^{5}z^{3}+12a^{5}z-3a^{5}z^{-1}-a^{4}z^{6}+6a^{4}z^{4}-9a^{4}z^{2}+3a^{4}+a^{3}z-a^{3}z^{-1}+a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

0

-4

1

0

-6

1

2

1

-8

1

-10

1

2

1

0

-12

0

-14

1

0

-16

1

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-8$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$
$r=-6$		${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$