L11n320

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_9,18,10,19 X_4,17,1,18 X_19,13,20,22 X_13,10,14,11 X_21,5,22,12 X_11,21,12,20
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, -4, -5, 7, 6, -8, 11, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(1)t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+2t(1)t(3)^{2}t(2)^{2}-2t(3)t(2)+t(2)+t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q+2-2q^{-1}+3q^{-2}-2q^{-3}+4q^{-4}-2q^{-5}+2q^{-6}-q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+2a^{8}-a^{6}z^{6}-6a^{6}z^{4}-11a^{6}z^{2}-2a^{6}z^{-2}-9a^{6}+a^{4}z^{8}+7a^{4}z^{6}+17a^{4}z^{4}+19a^{4}z^{2}+a^{4}z^{-2}+9a^{4}-a^{2}z^{6}-5a^{2}z^{4}-6a^{2}z^{2}-2a^{2}$ (db)
Kauffman polynomial	$z^{2}a^{10}-a^{10}+z^{3}a^{9}-za^{9}+z^{4}a^{8}-2z^{2}a^{8}-a^{8}z^{-2}+3a^{8}+z^{7}a^{7}-5z^{5}a^{7}+9z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}+2z^{8}a^{6}-12z^{6}a^{6}+25z^{4}a^{6}-25z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+z^{9}a^{5}-3z^{7}a^{5}-5z^{5}a^{5}+17z^{3}a^{5}-12za^{5}+2a^{5}z^{-1}+4z^{8}a^{4}-23z^{6}a^{4}+41z^{4}a^{4}-32z^{2}a^{4}-a^{4}z^{-2}+11a^{4}+z^{9}a^{3}-3z^{7}a^{3}-5z^{5}a^{3}+15z^{3}a^{3}-7za^{3}+2z^{8}a^{2}-11z^{6}a^{2}+17z^{4}a^{2}-10z^{2}a^{2}+3a^{2}+z^{7}a-5z^{5}a+6z^{3}a-2za$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

3

1

-1

1

-1

1

0

-3

1

3

1

-5

1

2

1

-7

4

2

-9

2

-11

1

4

3

0

-13

1

2

1

-15

1

0

-17

1

Integral Khovanov Homology

(db, data source)

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