L10a157

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,12,19,11 X_10,4,11,3 X_2,20,3,19 X_16,8,17,7 X_20,9,13,10 X_12,18,7,17 X_4,13,5,14 X_14,5,15,6 X_6,15,1,16
Gauss code	{1, -4, 3, -8, 9, -10}, {5, -1, 6, -3, 2, -7}, {8, -9, 10, -5, 7, -2, 4, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(u^{2}v^{2}w+u^{2}vw^{2}-u^{2}vw-u^{2}w^{2}-uv^{2}w+uv^{2}-uvw^{2}+uvw-uv+uw^{2}-uw-v^{2}-vw+v+w\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{-6}+3q^{-5}-q^{4}-5q^{-4}+3q^{3}+8q^{-3}-4q^{2}-9q^{-2}+8q+10q^{-1}-8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{2}z^{6}+z^{6}-a^{4}z^{4}+3a^{2}z^{4}-z^{4}a^{-2}+3z^{4}-2a^{4}z^{2}+2a^{2}z^{2}-2z^{2}a^{-2}+z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{3}+3a^{6}z^{4}-a^{6}z^{2}+5a^{5}z^{5}-3a^{5}z^{3}+7a^{4}z^{6}-10a^{4}z^{4}+4a^{4}z^{2}+7a^{3}z^{7}+z^{7}a^{-3}-13a^{3}z^{5}-4z^{5}a^{-3}+5a^{3}z^{3}+4z^{3}a^{-3}+5a^{2}z^{8}+3z^{8}a^{-2}-9a^{2}z^{6}-14z^{6}a^{-2}-a^{2}z^{4}+20z^{4}a^{-2}+3a^{2}z^{2}-8z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+2az^{9}+2z^{9}a^{-1}+az^{7}-5z^{7}a^{-1}-16az^{5}-2z^{5}a^{-1}+11az^{3}+6z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+8z^{8}-30z^{6}+32z^{4}-10z^{2}+2z^{-2}-3$ (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

9

1

-1

7

2

5

2

1

-1

3

6

2

4

1

3

0

-1

7

5

2

-3

4

5

1

-5

4

5

-1

-7

2

5

3

-9

1

3

-2

-11

2

-13

1

-1

Integral Khovanov Homology

(db, data source)

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