L10a55

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+4u^{2}v^{2}-u^{2}v-uv^{4}+5uv^{3}-9uv^{2}+5uv-u-v^{3}+4v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+12{\sqrt {q}}-{\frac {14}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-2a^{3}z^{3}+z^{3}a^{-3}+2za^{-3}+a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+5az^{3}-6z^{3}a^{-1}+az-4za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-6a^{2}z^{8}-4z^{8}a^{-2}-10z^{8}-8a^{3}z^{7}-8az^{7}-3z^{7}a^{-1}-3z^{7}a^{-3}-7a^{4}z^{6}+5a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+23z^{6}-4a^{5}z^{5}+9a^{3}z^{5}+23az^{5}+19z^{5}a^{-1}+9z^{5}a^{-3}-a^{6}z^{4}+7a^{4}z^{4}+2a^{2}z^{4}-6z^{4}a^{-2}+3z^{4}a^{-4}-15z^{4}+3a^{5}z^{3}-a^{3}z^{3}-14az^{3}-18z^{3}a^{-1}-8z^{3}a^{-3}-a^{4}z^{2}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+3z^{2}-2a^{3}z+5za^{-1}+3za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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

2

6

4

1

-3

4

6

2

4

2

6

4

-2

0

8

6

2

-2

6

7

1

-4

5

7

-2

-6

3

7

4

-8

1

4

-3

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$