L11n22

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+4t(1)t(2)^{3}+4t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {3}{q^{9/2}}}-{\frac {6}{q^{7/2}}}-q^{5/2}+{\frac {6}{q^{5/2}}}+2q^{3/2}-{\frac {5}{q^{3/2}}}+{\frac {1}{q^{13/2}}}-{\frac {3}{q^{11/2}}}-4{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}+z^{5}a^{5}+4z^{3}a^{5}+3za^{5}+2a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-8z^{3}a^{3}-8za^{3}-4a^{3}z^{-1}+2z^{5}a+8z^{3}a+7za+3az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+3a^{7}z^{3}-3a^{7}z+a^{6}z^{6}-a^{6}z^{4}+a^{6}z^{2}+a^{6}+3a^{5}z^{7}-12a^{5}z^{5}+20a^{5}z^{3}-15a^{5}z+2a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+11a^{4}z^{4}-4a^{4}z^{2}+2a^{4}+a^{3}z^{9}+2a^{3}z^{7}-23a^{3}z^{5}+37a^{3}z^{3}-23a^{3}z+4a^{3}z^{-1}+5a^{2}z^{8}-21a^{2}z^{6}+23a^{2}z^{4}-9a^{2}z^{2}+3a^{2}+az^{9}+z^{7}a^{-1}-16az^{5}-5z^{5}a^{-1}+28az^{3}+8z^{3}a^{-1}-16az-5za^{-1}+3az^{-1}+a^{-1}z^{-1}+2z^{8}-9z^{6}+11z^{4}-5z^{2}+1$ (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

χ

6

1

4

1

-1

2

3

1

2

0

2

1

-1

-2

1

4

3

0

-4

4

3

-1

-6

3

0

-8

1

2

4

3

-10

2

0

-12

2

-14

1

-1

Integral Khovanov Homology

(db, data source)

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