L10a47

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_18,8,19,7 X_20,10,5,9 X_8,20,9,19 X_16,12,17,11 X_12,16,13,15 X_10,18,11,17 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -5, 4, -8, 6, -7, 10, -2, 7, -6, 8, -3, 5, -4}

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

3

1

2

8

4

2

-2

6

4

3

1

4

5

4

-1

2

3

4

-1

0

3

6

3

-2

1

2

-1

-4

3

-6

1

0

-8

1

Integral Khovanov Homology

(db, data source)

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