L11n378

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_5,10,6,11 X₃₈₄₉ X_22,18,19,17 X_11,20,12,21 X_19,12,20,13 X_18,22,5,21 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -11, -5, 3}, {-8, 7, 9, -6}, {-4, -1, 2, 5, -10, 4, -7, 8, 11, -2, -3, 10, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$-q^{-7}+q^{-6}-2q^{-5}+3q^{-4}-2q^{-3}+3q^{-2}-q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}z^{-2}-2a^{6}+z^{4}a^{4}+4z^{2}a^{4}+4a^{4}z^{-2}+7a^{4}-3z^{2}a^{2}-5a^{2}z^{-2}-8a^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+11a^{7}z^{3}-6a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+7a^{6}z^{4}-4a^{6}z^{2}-a^{6}z^{-2}+3a^{6}+3a^{5}z^{7}-16a^{5}z^{5}+27a^{5}z^{3}-19a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+6a^{4}z^{4}-11a^{4}z^{2}-4a^{4}z^{-2}+10a^{4}+2a^{3}z^{7}-10a^{3}z^{5}+19a^{3}z^{3}-21a^{3}z+9a^{3}z^{-1}+a^{2}z^{6}-a^{2}z^{4}-7a^{2}z^{2}-5a^{2}z^{-2}+11a^{2}+3az^{3}-8az+5az^{-1}-2z^{-2}+5$ (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

χ

1

3

-1

2

4

2

-3

1

2

-5

1

2

1

-7

2

1

-9

1

-11

1

2

-1

-13

0

-15

1

-1

Integral Khovanov Homology

(db, data source)

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