L11n390

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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_5,12,6,13 X₃₈₄₉ X_22,14,19,13 X_20,10,21,9 X_10,20,11,19 X_14,22,15,21 X_11,18,12,5 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, 7, -8, -10, 4, 6, -9, -11, -2, 3, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)\left(t(1)t(3)^{4}+t(1)t(3)^{3}-t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}+t(3)^{2}+t(2)t(3)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-7}-3q^{-6}+3q^{-5}-3q^{-4}+4q^{-3}+q^{2}-2q^{-2}+2q^{-1}+1$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}-a^{6}z^{-2}-a^{6}+4a^{4}z^{2}+4a^{4}z^{-2}+8a^{4}-a^{2}z^{6}-7a^{2}z^{4}-14a^{2}z^{2}-5a^{2}z^{-2}-13a^{2}+z^{4}+5z^{2}+2z^{-2}+6$ (db)
Kauffman polynomial	$z^{4}a^{8}-z^{2}a^{8}+3z^{5}a^{7}-6z^{3}a^{7}+a^{7}z^{-1}+2z^{6}a^{6}-3z^{4}a^{6}-a^{6}z^{-2}+a^{6}+3z^{5}a^{5}-9za^{5}+5a^{5}z^{-1}+6z^{4}a^{4}-13z^{2}a^{4}-4a^{4}z^{-2}+12a^{4}+z^{7}a^{3}-9z^{5}a^{3}+28z^{3}a^{3}-27za^{3}+9a^{3}z^{-1}+z^{8}a^{2}-10z^{6}a^{2}+31z^{4}a^{2}-37z^{2}a^{2}-5a^{2}z^{-2}+21a^{2}+z^{7}a-9z^{5}a+22z^{3}a-18za+5az^{-1}+z^{8}-8z^{6}+21z^{4}-23z^{2}-2z^{-2}+11$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

5

1

3

1

-1

3

-3

2

3

1

0

-5

3

1

2

-7

2

1

-9

3

0

-11

1

0

-13

2

-2

-15

1

Integral Khovanov Homology

(db, data source)

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