L11n421

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_3,10,4,11 X_17,2,18,3 X_9,19,10,18 X_20,12,21,11 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13 X_4,19,5,20 X_12,22,7,21
Gauss code	{1, 4, -3, -10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, -4, 5, 10, -6, 11, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{2}t(3)^{3}-t(2)^{2}t(3)^{3}-2t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}+t(1)t(2)t(3)^{2}-t(1)^{2}t(3)+2t(1)t(3)-t(1)t(2)t(3)+t(1)^{2}-t(1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{5}-q^{4}+3q^{3}-3q^{2}+4q-3+4q^{-1}-2q^{-2}+2q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-3a^{2}z^{2}-8z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}-8a^{-2}+z^{6}+5z^{4}+8z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$z^{2}a^{-6}-a^{-6}+z^{3}a^{-5}+2z^{4}a^{-4}-4z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+a^{3}z^{7}+z^{7}a^{-3}-5a^{3}z^{5}-4z^{5}a^{-3}+6a^{3}z^{3}+7z^{3}a^{-3}-a^{3}z-6za^{-3}+2a^{-3}z^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}-11a^{2}z^{6}-11z^{6}a^{-2}+18a^{2}z^{4}+23z^{4}a^{-2}-11a^{2}z^{2}-25z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+12a^{-2}+az^{9}+z^{9}a^{-1}-3az^{7}-3z^{7}a^{-1}-3az^{5}-2z^{5}a^{-1}+9az^{3}+9z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-22z^{6}+39z^{4}-31z^{2}-z^{-2}+10$ (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

χ

11

1

9

1

0

7

2

5

2

0

3

2

1

2

3

1

-1

2

1

-3

1

3

2

-5

1

0

-7

1

-9

1

-1

Integral Khovanov Homology

(db, data source)

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