L11n269

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,17,8,16 X_15,5,16,8 X_18,11,19,12 X_22,17,9,18 X_20,13,21,14 X_12,19,13,20 X_14,21,15,22 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -8, 7, -9, -4, 3, 6, -5, 8, -7, 9, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(3)^{5}-t(1)t(3)^{4}-t(2)t(3)^{4}+t(1)t(3)^{3}+t(2)t(3)^{3}-t(1)t(3)^{2}-t(2)t(3)^{2}+t(1)t(3)+t(2)t(3)-t(1)t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{-8}+q^{-7}-2q^{-6}+2q^{-5}-3q^{-4}+3q^{-3}+q^{2}+2q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{4}\right)-4a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{6}+7a^{4}z^{4}+18a^{4}z^{2}+7a^{4}z^{-2}+18a^{4}-a^{2}z^{6}-8a^{2}z^{4}-20a^{2}z^{2}-8a^{2}z^{-2}-20a^{2}+z^{4}+5z^{2}+3z^{-2}+7$ (db)
Kauffman polynomial	$a^{9}z^{5}-4a^{9}z^{3}+3a^{9}z-a^{9}z^{-1}+a^{8}z^{6}-3a^{8}z^{4}+a^{8}+a^{7}z^{7}-3a^{7}z^{5}+3a^{7}z-a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+9a^{6}z^{4}-9a^{6}z^{2}-2a^{6}z^{-2}+7a^{6}+2a^{5}z^{7}-12a^{5}z^{5}+26a^{5}z^{3}-21a^{5}z+7a^{5}z^{-1}+a^{4}z^{8}-8a^{4}z^{6}+24a^{4}z^{4}-32a^{4}z^{2}-7a^{4}z^{-2}+22a^{4}+2a^{3}z^{7}-17a^{3}z^{5}+46a^{3}z^{3}-45a^{3}z+15a^{3}z^{-1}+a^{2}z^{8}-10a^{2}z^{6}+33a^{2}z^{4}-46a^{2}z^{2}-8a^{2}z^{-2}+28a^{2}+az^{7}-9az^{5}+24az^{3}-24az+8az^{-1}+z^{8}-8z^{6}+21z^{4}-23z^{2}-3z^{-2}+13$ (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

1

-1

3

-3

1

4

1

2

-5

3

-7

1

2

1

0

-9

2

3

-1

-11

1

0

-13

1

2

-1

-15

0

-17

1

-1

Integral Khovanov Homology

(db, data source)

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