L11n198

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_5,14,6,15 X_7,17,8,16 X_15,21,16,20 X_18,14,19,13 X_21,6,22,7 X_22,18,9,17 X_19,5,20,4 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, 9, -3, 7, -4, -11}, {10, -1, 11, -2, 6, 3, -5, 4, 8, -6, -9, 5, -7, -8}

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

Polynomial invariants

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

-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

3

2

1

-2

2

3

1

-4

1

2

-1

-6

2

-8

1

0

-10

1

Integral Khovanov Homology

(db, data source)

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