L11n429

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,20,10,21 X_14,5,15,6 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_3,10,4,11 X_18,11,19,12 X_6,15,1,16 X_19,4,20,5 X_2,21,3,22
Gauss code	{1, -11, -7, 10, 3, -9}, {5, -1, -2, 7, 8, -4}, {4, -3, 9, -5, 6, -8, -10, 2, 11, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}+u^{2}v^{2}w-u^{2}vw^{3}+2u^{2}vw^{2}+uv^{2}w^{2}-uv^{2}w-uvw^{2}+uvw+uw^{2}-uw-2vw+v-w^{2}+2w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q+3-4q^{-1}+6q^{-2}-6q^{-3}+7q^{-4}-5q^{-5}+5q^{-6}-2q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+a^{8}-a^{6}z^{6}-5a^{6}z^{4}-7a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{8}+6a^{4}z^{6}+12a^{4}z^{4}+11a^{4}z^{2}+a^{4}z^{-2}+4a^{4}-a^{2}z^{6}-4a^{2}z^{4}-3a^{2}z^{2}$ (db)
Kauffman polynomial	$a^{10}z^{2}+2a^{9}z^{3}+5a^{8}z^{4}-9a^{8}z^{2}-a^{8}z^{-2}+6a^{8}+2a^{7}z^{7}-5a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-18a^{6}z^{6}+31a^{6}z^{4}-29a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+2a^{5}z^{9}-5a^{5}z^{7}-3a^{5}z^{5}+10a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+7a^{4}z^{8}-32a^{4}z^{6}+44a^{4}z^{4}-26a^{4}z^{2}-a^{4}z^{-2}+8a^{4}+2a^{3}z^{9}-6a^{3}z^{7}-2a^{3}z^{5}+10a^{3}z^{3}-3a^{3}z+3a^{2}z^{8}-14a^{2}z^{6}+18a^{2}z^{4}-7a^{2}z^{2}+a^{2}+az^{7}-4az^{5}+4az^{3}-az$ (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

χ

3

1

-1

1

2

-1

2

1

-1

-3

4

2

-5

3

0

-7

4

3

1

-9

2

4

2

-11

3

0

-13

3

-15

1

2

-1

-17

1

Integral Khovanov Homology

(db, data source)

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