L11n162

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_17,13,18,12 X_14,5,15,6 X_4,13,5,14 X_11,19,12,18 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 9, -3, 6, -7, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}+t(2)^{4}+t(1)^{2}t(2)^{3}-2t(1)t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+t(2)+t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+3q^{5/2}-{\frac {2}{q^{5/2}}}-4q^{3/2}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{11/2}}}+4{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{5}+a^{5}z^{-1}-z^{5}a-4z^{3}a-5za-2az^{-1}-z^{5}a^{-1}-3z^{3}a^{-1}-za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}+2za^{-3}$ (db)
Kauffman polynomial	$a^{5}z^{7}-7a^{5}z^{5}+14a^{5}z^{3}-9a^{5}z+a^{5}z^{-1}+z^{6}a^{-4}-2a^{4}z^{4}-4z^{4}a^{-4}+4a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+2z^{7}a^{-3}-2a^{3}z^{5}-8z^{5}a^{-3}+6a^{3}z^{3}+7z^{3}a^{-3}-3a^{3}z-2za^{-3}+a^{2}z^{8}+2z^{8}a^{-2}-7a^{2}z^{6}-8z^{6}a^{-2}+15a^{2}z^{4}+8z^{4}a^{-2}-10a^{2}z^{2}-5z^{2}a^{-2}+3a^{2}+2a^{-2}+az^{9}+z^{9}a^{-1}-6az^{7}-3z^{7}a^{-1}+13az^{5}-13az^{3}+2z^{3}a^{-1}+8az-2az^{-1}-a^{-1}z^{-1}+3z^{8}-16z^{6}+29z^{4}-22z^{2}+5$ (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

4

5

χ

10

1

-1

8

1

6

2

1

-1

4

2

1

2

0

1

3

2

0

-2

2

3

1

-4

1

2

-1

-6

2

-8

1

0

-10

1

-12

1

Integral Khovanov Homology

(db, data source)

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