L11n228

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_17,9,18,22 X_9,21,10,20 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_19,5,20,4 X_5,19,6,18 X_21,17,22,16
Gauss code	{1, -2, 3, 9, -10, -6, 7, -8}, {-5, -1, 2, -3, 6, -7, 8, 11, -4, 10, -9, 5, -11, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{3}-u^{3}v^{2}+u^{2}v^{3}-u^{2}v^{2}+2u^{2}v-u^{2}-uv^{3}+2uv^{2}-uv+u-v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+q^{7/2}+{\frac {3}{q^{7/2}}}-q^{5/2}-{\frac {3}{q^{5/2}}}+q^{3/2}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+2a^{3}z^{3}+4a^{3}z+2a^{3}z^{-1}-az^{7}-7az^{5}+z^{5}a^{-1}-15az^{3}+5z^{3}a^{-1}-13az+6za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-z^{8}a^{-2}-2z^{8}-a^{5}z^{7}+8az^{7}+7z^{7}a^{-1}-2a^{6}z^{6}-2a^{4}z^{6}+8a^{2}z^{6}+7z^{6}a^{-2}+15z^{6}-a^{7}z^{5}+a^{5}z^{5}-2a^{3}z^{5}-19az^{5}-15z^{5}a^{-1}+6a^{6}z^{4}+4a^{4}z^{4}-18a^{2}z^{4}-15z^{4}a^{-2}-31z^{4}+3a^{7}z^{3}+3a^{5}z^{3}+6a^{3}z^{3}+20az^{3}+14z^{3}a^{-1}-3a^{6}z^{2}+12a^{2}z^{2}+11z^{2}a^{-2}+20z^{2}-2a^{7}z-6a^{3}z-15az-7za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

0

4

1

0

2

1

0

2

1

3

-2

2

3

1

0

-4

2

1

-6

1

2

1

0

-8

2

0

-10

1

2

1

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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