L10a68

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_16,13,17,14 X_14,6,15,5 X_4,16,5,15 X_20,18,7,17 X_18,12,19,11 X_12,20,13,19
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-5uv^{2}+3uv-u-v^{3}+3v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-2q^{7/2}+{\frac {5}{q^{7/2}}}+4q^{5/2}-{\frac {8}{q^{5/2}}}-7q^{3/2}+{\frac {9}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+8{\sqrt {q}}-{\frac {10}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z+3za^{-3}+a^{-3}z^{-1}+az^{7}+5az^{5}-2z^{5}a^{-1}+9az^{3}-8z^{3}a^{-1}+7az-9za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-2z^{8}a^{-2}-5z^{8}-4a^{3}z^{7}-4az^{7}-2z^{7}a^{-1}-2z^{7}a^{-3}-4a^{4}z^{6}+3a^{2}z^{6}+4z^{6}a^{-2}-z^{6}a^{-4}+12z^{6}-3a^{5}z^{5}+4a^{3}z^{5}+14az^{5}+14z^{5}a^{-1}+7z^{5}a^{-3}-a^{6}z^{4}+4a^{4}z^{4}-a^{2}z^{4}+3z^{4}a^{-2}+4z^{4}a^{-4}-7z^{4}+4a^{5}z^{3}-a^{3}z^{3}-17az^{3}-19z^{3}a^{-1}-7z^{3}a^{-3}+a^{6}z^{2}-a^{2}z^{2}-7z^{2}a^{-2}-4z^{2}a^{-4}-3z^{2}-a^{5}z+9az+12za^{-1}+4za^{-3}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

3

1

-2

4

1

3

2

4

3

-1

0

6

4

2

-2

4

5

1

-4

4

5

-1

-6

2

5

3

-8

1

3

-2

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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