L11n324

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_11,21,12,20 X_19,22,20,13 X_13,12,14,5 X_17,1,18,4 X_21,11,22,10
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11, -7, 9}, {-9, 2, 4, -5, -10, 6, -8, 7, -11, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(3)^{2}t(2)^{3}+t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-2t(1)t(3)^{2}t(2)^{2}+2t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+2t(1)t(3)^{2}t(2)-t(3)^{2}t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)t(3)^{2}+t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-8}+2q^{-7}-3q^{-6}+6q^{-5}-7q^{-4}+8q^{-3}-6q^{-2}+2q+6q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{6}-3z^{2}a^{6}-2a^{6}+z^{6}a^{4}+5z^{4}a^{4}+10z^{2}a^{4}+a^{4}z^{-2}+7a^{4}-3z^{4}a^{2}-10z^{2}a^{2}-2a^{2}z^{-2}-9a^{2}+2z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{9}z^{5}-3a^{9}z^{3}+a^{9}z+2a^{8}z^{6}-6a^{8}z^{4}+4a^{8}z^{2}-a^{8}+2a^{7}z^{7}-4a^{7}z^{5}+a^{7}z+2a^{6}z^{8}-5a^{6}z^{6}+4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{9}-a^{5}z^{7}-2a^{5}z^{5}+8a^{5}z^{3}-3a^{5}z+4a^{4}z^{8}-16a^{4}z^{6}+32a^{4}z^{4}-26a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+a^{3}z^{9}-2a^{3}z^{7}+2a^{3}z^{5}+6a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}+2a^{2}z^{8}-9a^{2}z^{6}+25a^{2}z^{4}-31a^{2}z^{2}-2a^{2}z^{-2}+13a^{2}+az^{7}-az^{5}+az^{3}-5az+2az^{-1}+3z^{4}-8z^{2}-z^{-2}+6$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

2

1

-1

5

2

3

-3

4

0

-5

4

2

-7

3

4

1

-9

3

4

-1

-11

1

4

3

-13

1

2

-1

-15

1

-17

1

-1

Integral Khovanov Homology

(db, data source)

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