L11n327

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_13,18,14,19 X_9,21,10,20 X_19,5,20,12 X_11,13,12,22 X_21,11,22,10 X_17,1,18,4
Gauss code	{1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, 4, -5, -11, 6, -8, 7, -10, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{3}-2uv^{2}w^{2}-uvw^{3}+3uvw^{2}-2uvw-uw^{2}+uw+v^{3}\left(-w^{2}\right)+v^{3}w+2v^{2}w^{2}-3v^{2}w+v^{2}+2vw-v}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$2q^{5}-4q^{4}+6q^{3}-6q^{2}+8q-6+6q^{-1}-3q^{-2}+2q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$2z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-a^{2}z^{4}-3z^{4}a^{-2}-3a^{2}z^{2}-10z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+z^{6}+5z^{4}+10z^{2}+z^{-2}+8$ (db)
Kauffman polynomial	$3z^{2}a^{-6}-a^{-6}+z^{5}a^{-5}+3z^{3}a^{-5}-za^{-5}+3z^{6}a^{-4}-3z^{4}a^{-4}-a^{-4}z^{-2}+3a^{-4}+a^{3}z^{7}+4z^{7}a^{-3}-5a^{3}z^{5}-10z^{5}a^{-3}+7a^{3}z^{3}+12z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}-10a^{2}z^{6}-8z^{6}a^{-2}+16a^{2}z^{4}+12z^{4}a^{-2}-10a^{2}z^{2}-19z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+az^{9}+z^{9}a^{-1}-az^{7}+2z^{7}a^{-1}-9az^{5}-15z^{5}a^{-1}+16az^{3}+18z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+5z^{8}-21z^{6}+31z^{4}-26z^{2}-z^{-2}+11$ (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

χ

11

2

9

3

1

-2

7

3

1

2

5

3

0

3

5

3

2

1

4

6

2

-1

2

0

-3

1

4

3

-5

1

2

-1

-7

1

-9

1

-1

Integral Khovanov Homology

(db, data source)

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