L11n247

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X₃₈₄₉ X_14,6,15,5 X_7,18,8,19 X_9,21,10,20 X_10,11,1,12 X_6,14,7,13 X_17,4,18,5 X_15,11,16,22 X_19,3,20,2 X_21,17,22,16
Gauss code	{1, 10, -2, 8, 3, -7, -4, 2, -5, -6}, {6, -1, 7, -3, -9, 11, -8, 4, -10, 5, -11, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z+2za^{-3}+az^{5}-z^{5}a^{-1}+5az^{3}-5z^{3}a^{-1}+6az-6za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{4}z^{6}+z^{6}a^{-4}-4a^{4}z^{4}-4z^{4}a^{-4}+2a^{4}z^{2}+2z^{2}a^{-4}+2a^{3}z^{7}+2z^{7}a^{-3}-10a^{3}z^{5}-10z^{5}a^{-3}+11a^{3}z^{3}+11z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{2}z^{8}+z^{8}a^{-2}-5a^{2}z^{6}-5z^{6}a^{-2}+4a^{2}z^{4}+4z^{4}a^{-2}+3az^{7}+3z^{7}a^{-1}-19az^{5}-19z^{5}a^{-1}+29az^{3}+29z^{3}a^{-1}-12az-12za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{8}-12z^{6}+16z^{4}-4z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

1

4

1

2

1

0

2

1

0

2

4

2

0

-2

1

2

1

-4

1

2

1

0

-6

1

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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