L10n63

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_7,17,8,16 X₃₉₄₈ X_17,2,18,3 X_5,14,6,15 X_11,6,12,7 X_9,18,10,19 X_15,11,16,20 X_10,13,1,14 X_19,5,20,4
Gauss code	{1, 4, -3, 10, -5, 6, -2, 3, -7, -9}, {-6, -1, 9, 5, -8, 2, -4, 7, -10, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(uv+1)\left(u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+3uv-2u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-7q^{3/2}+9{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {10}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{8}-3z^{8}-6a^{3}z^{7}-12az^{7}-6z^{7}a^{-1}-3a^{4}z^{6}-a^{2}z^{6}-4z^{6}a^{-2}-2z^{6}+13a^{3}z^{5}+26az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+6a^{2}z^{4}+7z^{4}a^{-2}+13z^{4}-6a^{5}z^{3}-18a^{3}z^{3}-18az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-2z^{2}a^{-2}-5z^{2}+6a^{5}z+9a^{3}z+4az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

1

3

2

5

3

-2

0

6

4

2

-2

5

6

1

-4

4

5

-1

-6

2

5

3

-8

1

4

-3

-10

3

Integral Khovanov Homology

(db, data source)

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