L11n157

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_17,15,18,14 X_7,17,8,16 X_15,7,16,22 X_13,19,14,18 X_6,20,1,19 X_20,12,21,11 X_12,6,13,5 X_4,21,5,22
Gauss code	{1, -2, 3, -11, 10, -8}, {-5, -1, 2, -3, 9, -10, -7, 4, -6, 5, -4, 7, 8, -9, 11, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-2t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+7q^{7/2}-{\frac {1}{q^{7/2}}}-10q^{5/2}+{\frac {3}{q^{5/2}}}+10q^{3/2}-{\frac {6}{q^{3/2}}}+2q^{11/2}-11{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-5z^{5}a^{-1}+z^{5}a^{-3}+3az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+3az-5za^{-1}+za^{-5}+az^{-1}-2a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$3z^{4}a^{-6}-6z^{2}a^{-6}+2a^{-6}+z^{7}a^{-5}+z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-4z^{6}a^{-4}+10z^{4}a^{-4}-12z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}+2z^{7}a^{-3}+a^{3}z^{5}-5z^{5}a^{-3}-2a^{3}z^{3}+6z^{3}a^{-3}+a^{3}z-2a^{-3}z^{-1}+5z^{8}a^{-2}+3a^{2}z^{6}-9z^{6}a^{-2}-6a^{2}z^{4}+9z^{4}a^{-2}+2a^{2}z^{2}-5z^{2}a^{-2}+3a^{-2}+z^{9}a^{-1}+4az^{7}+5z^{7}a^{-1}-7az^{5}-14z^{5}a^{-1}+3az^{3}+14z^{3}a^{-1}-3az-5za^{-1}+az^{-1}+3z^{8}-2z^{6}-4z^{4}+3z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

2

-2

10

2

8

5

2

-3

6

5

2

3

4

5

0

2

6

5

1

0

3

6

3

-2

3

5

-2

-4

1

4

3

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

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