L11n287

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_4,15,1,16 X_21,18,22,19 X_9,21,10,20 X_19,5,20,10 X_11,16,12,17 X_17,22,18,11
Gauss code	{1, -4, 3, -6}, {-2, -1, 5, -3, -8, 9}, {-10, 2, 4, -5, 6, 10, -11, 7, -9, 8, -7, 11}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$q^{2}+2-q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-2q^{-5}+2q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}-a^{2}z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{-2}-z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+3a^{4}z^{8}+a^{2}z^{8}+a^{7}z^{7}-4a^{5}z^{7}-5a^{3}z^{7}-11a^{6}z^{6}-17a^{4}z^{6}-6a^{2}z^{6}-5a^{7}z^{5}+6a^{3}z^{5}+az^{5}+16a^{6}z^{4}+28a^{4}z^{4}+12a^{2}z^{4}+6a^{7}z^{3}+8a^{5}z^{3}-2a^{3}z^{3}-4az^{3}-7a^{6}z^{2}-18a^{4}z^{2}-10a^{2}z^{2}+z^{2}a^{-2}+2z^{2}-2a^{7}z-4a^{5}z+4az+2za^{-1}+2a^{6}+4a^{4}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

3

1

2

-1

2

4

1

-3

2

-5

2

3

1

-7

1

2

1

0

-9

1

2

1

0

-11

1

0

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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