L11n333

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_11,16,12,17 X₃₈₄₉ X_2,18,3,17 X_5,14,6,15 X_18,7,19,8 X_15,12,16,5 X_13,20,14,21 X_9,13,10,22 X_21,11,22,10 X_19,1,20,4
Gauss code	{1, -4, -3, 11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, 4, -6, -11, 8, -10, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v+w-1)(vw-v-w)(v-uw)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-3q^{-5}+6q^{-4}-5q^{-3}+q^{2}+6q^{-2}-2q-5q^{-1}+5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+z^{4}a^{4}+2z^{2}a^{4}+a^{4}z^{-2}+3a^{4}+z^{4}a^{2}-2a^{2}z^{-2}-3a^{2}-2z^{2}+z^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+5a^{4}z^{8}+3a^{2}z^{8}+a^{7}z^{7}-a^{5}z^{7}+a^{3}z^{7}+3az^{7}-10a^{6}z^{6}-23a^{4}z^{6}-12a^{2}z^{6}+z^{6}-5a^{7}z^{5}-10a^{5}z^{5}-16a^{3}z^{5}-11az^{5}+16a^{6}z^{4}+36a^{4}z^{4}+19a^{2}z^{4}-z^{4}+7a^{7}z^{3}+17a^{5}z^{3}+23a^{3}z^{3}+15az^{3}+2z^{3}a^{-1}-11a^{6}z^{2}-31a^{4}z^{2}-20a^{2}z^{2}+z^{2}a^{-2}+z^{2}-2a^{7}z-7a^{5}z-12a^{3}z-8az-za^{-1}+3a^{6}+11a^{4}+11a^{2}-a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

5

1

3

1

-1

1

4

1

3

-1

4

0

-3

2

1

-5

3

4

1

-7

3

2

1

-9

1

4

3

-11

1

2

-1

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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