L10n73

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,17,8,16 X_20,9,11,10 X_18,12,19,11 X_15,9,16,8 X_10,19,5,20 X_17,14,18,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 6, 4, -7}, {5, -2, 10, 8, -6, 3, -8, -5, 7, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v+w-1)(vw-v-w)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-4q^{3}-q^{-3}+6q^{2}+4q^{-2}-6q-4q^{-1}+7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-2z^{3}a^{-5}+3z^{6}a^{-4}-8z^{4}a^{-4}+4z^{2}a^{-4}+3z^{7}a^{-3}-7z^{5}a^{-3}+a^{3}z^{3}+3z^{3}a^{-3}+z^{8}a^{-2}+2z^{6}a^{-2}+4a^{2}z^{4}-8z^{4}a^{-2}-3a^{2}z^{2}+5z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+az^{7}+4z^{7}a^{-1}-8z^{5}a^{-1}+az^{3}+5z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+z^{8}-z^{6}+4z^{4}-2z^{2}+2z^{-2}-3$ (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		\

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

2

1

-1

5

4

2

3

0

1

4

3

1

-1

2

5

3

-3

2

0

-5

3

-7

1

-1

Integral Khovanov Homology

(db, data source)

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