L11n179

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_3,10,4,11 X_16,7,17,8 X_20,10,21,9 X_22,15,7,16 X_14,6,15,5 X_18,14,19,13 X_12,20,13,19 X_11,4,12,5 X_6,18,1,17 X_2,21,3,22
Gauss code	{1, -11, -2, 9, 6, -10}, {3, -1, 4, 2, -9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}-t(2)^{4}+t(1)^{2}t(2)^{3}-4t(1)t(2)^{3}+2t(2)^{3}-2t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-2t(2)^{2}+2t(1)^{2}t(2)-4t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {8}{q^{7/2}}}+q^{5/2}-{\frac {10}{q^{5/2}}}-4q^{3/2}+{\frac {10}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+6{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{5}a^{3}-2z^{3}a^{3}-3za^{3}-2a^{3}z^{-1}-z^{5}a-z^{3}a+za+2az^{-1}+z^{3}a^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-4a^{4}z^{8}-5a^{2}z^{8}-z^{8}-5a^{5}z^{7}-5a^{3}z^{7}-3a^{6}z^{6}+9a^{4}z^{6}+12a^{2}z^{6}-a^{7}z^{5}+13a^{5}z^{5}+18a^{3}z^{5}-4z^{5}a^{-1}+6a^{6}z^{4}-8a^{4}z^{4}-15a^{2}z^{4}-z^{4}a^{-2}-2z^{4}+2a^{7}z^{3}-11a^{5}z^{3}-23a^{3}z^{3}-5az^{3}+5z^{3}a^{-1}-a^{6}z^{2}+3a^{4}z^{2}+6a^{2}z^{2}+z^{2}a^{-2}+3z^{2}+5a^{5}z+11a^{3}z+7az+za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

3

1

-2

0

6

3

-2

5

4

-1

-4

5

0

-6

4

6

2

-8

2

4

-2

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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