L11n436

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,8,19,7 X_5,16,6,17 X_15,10,16,11 X_3,15,4,14 X_11,5,12,4 X_2,20,3,19 X_20,9,21,10 X_13,7,14,12 X_17,22,18,13 X_6,21,1,22
Gauss code	{1, -7, -5, 6, -3, -11}, {2, -1, 8, 4, -6, 9}, {-9, 5, -4, 3, -10, -2, 7, -8, 11, 10}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)(w+1)^{2}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{4}+2q^{3}-q^{2}+2q+1+q^{-2}-q^{-3}+2q^{-4}-q^{-5}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}-z^{4}a^{-2}-a^{2}z^{2}-3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+z^{6}+5z^{4}+5z^{2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}-3a^{5}z^{3}+2a^{4}z^{6}-8a^{4}z^{4}+4a^{4}z^{2}+a^{3}z^{7}+z^{7}a^{-3}-5a^{3}z^{5}-5z^{5}a^{-3}+4a^{3}z^{3}+5z^{3}a^{-3}+a^{2}z^{8}+2z^{8}a^{-2}-6a^{2}z^{6}-12z^{6}a^{-2}+8a^{2}z^{4}+20z^{4}a^{-2}-4a^{2}z^{2}-12z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+az^{9}+z^{9}a^{-1}-5az^{7}-5z^{7}a^{-1}+2az^{5}+3z^{5}a^{-1}+6az^{3}+4z^{3}a^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-20z^{6}+36z^{4}-20z^{2}+2z^{-2}+1$ (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

5

χ

9

1

-1

7

1

5

1

3

2

1

4

2

1

3

-1

2

5

2

1

-3

1

2

1

-5

1

2

1

0

-7

1

-9

1

-11

1

-1

Integral Khovanov Homology

(db, data source)

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