L10a158

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(u^{2}v^{2}+u^{2}vw-u^{2}v-u^{2}w+uv^{2}w-uv^{2}+uvw^{2}-3uvw+uv-uw^{2}+uw-v^{2}w-vw^{2}+vw+w^{2}\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-5q^{3}+9q^{2}-10q+12-10q^{-1}+9q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}\left(-z^{2}\right)-z^{2}a^{-4}+a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{2}+a^{-2}+2z^{4}+z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+4a^{2}z^{8}+4z^{8}a^{-2}+8z^{8}+4a^{3}z^{7}-az^{7}-z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}-9a^{2}z^{6}-9z^{6}a^{-2}+3z^{6}a^{-4}-24z^{6}+a^{5}z^{5}-8a^{3}z^{5}-2az^{5}-2z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-7a^{4}z^{4}+9a^{2}z^{4}+9z^{4}a^{-2}-7z^{4}a^{-4}+32z^{4}-2a^{5}z^{3}+3a^{3}z^{3}+3az^{3}+3z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}-3a^{2}z^{2}-3z^{2}a^{-2}+3z^{2}a^{-4}-12z^{2}+2az+2za^{-1}-2a^{2}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

11

1

-1

9

2

7

3

1

-2

5

6

2

4

3

5

4

-1

1

7

5

2

-1

5

7

2

-3

4

5

-1

-5

2

6

4

-7

1

3

-2

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

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