L11n382

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_22,18,19,17 X_20,12,21,11 X_12,20,13,19 X_18,22,5,21 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, -10, 4, 7, -8, 11, -2, 3, 10, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)^{2}(t(2)+t(3))}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{4}-2q^{3}-2q^{-3}+6q^{2}+4q^{-2}-5q-5q^{-1}+6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2a^{-4}z^{-2}+2a^{-4}-2a^{2}z^{2}-4z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{2}-8a^{-2}+2z^{4}+6z^{2}+4z^{-2}+8$ (db)
Kauffman polynomial	$3z^{4}a^{-4}-9z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+z^{7}a^{-3}-3z^{5}a^{-3}+3a^{3}z^{3}+9z^{3}a^{-3}-3a^{3}z-13za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}-3z^{6}a^{-2}+2a^{2}z^{4}+11z^{4}a^{-2}-4a^{2}z^{2}-20z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{2}+16a^{-2}+2az^{7}+3z^{7}a^{-1}-5az^{5}-8z^{5}a^{-1}+13az^{3}+19z^{3}a^{-1}-14az-24za^{-1}+5az^{-1}+9a^{-1}z^{-1}+z^{8}-2z^{6}+10z^{4}-15z^{2}-4z^{-2}+11$ (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

χ

9

2

7

2

0

5

4

3

1

2

1

5

4

1

-1

2

3

1

-3

2

3

-1

-5

2

-7

2

-2

Integral Khovanov Homology

(db, data source)

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