L11n363

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_5,18,6,19 X₈₄₉₃ X_9,21,10,20 X_19,11,20,10 X_15,22,16,17 X_17,16,18,5 X_21,14,22,15 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {-9, 4, -7, 6, -10, 8}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10, -8, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}}{{\sqrt {t(1)}}t(3)}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-3q^{-6}+4q^{-5}-3q^{-4}+4q^{-3}+q^{2}-q^{-2}-q+q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-3a^{6}+z^{6}a^{4}+6z^{4}a^{4}+13z^{2}a^{4}+a^{4}z^{-2}+9a^{4}-z^{6}a^{2}-7z^{4}a^{2}-14z^{2}a^{2}-2a^{2}z^{-2}-10a^{2}+z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{5}a^{9}-3z^{3}a^{9}+za^{9}+2z^{6}a^{8}-6z^{4}a^{8}+3z^{2}a^{8}-a^{8}+2z^{7}a^{7}-6z^{5}a^{7}+3z^{3}a^{7}+z^{8}a^{6}-3z^{6}a^{6}+3z^{4}a^{6}-3z^{2}a^{6}+2a^{6}+2z^{7}a^{5}-9z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+2z^{8}a^{4}-15z^{6}a^{4}+37z^{4}a^{4}-34z^{2}a^{4}-a^{4}z^{-2}+13a^{4}+z^{9}a^{3}-7z^{7}a^{3}+11z^{5}a^{3}+3z^{3}a^{3}-9za^{3}+2a^{3}z^{-1}+2z^{8}a^{2}-17z^{6}a^{2}+43z^{4}a^{2}-41z^{2}a^{2}-2a^{2}z^{-2}+15a^{2}+z^{9}a-7z^{7}a+13z^{5}a-5z^{3}a-4za+2az^{-1}+z^{8}-7z^{6}+15z^{4}-13z^{2}-z^{-2}+6$ (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

3

4

χ

5

1

3

0

1

0

-1

3

1

2

-3

1

2

1

0

-5

4

2

1

3

-7

2

3

1

-9

2

1

-11

1

2

1

-13

1

2

-1

-15

1

-17

1

-1

Integral Khovanov Homology

(db, data source)

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