L11a370

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_20,8,21,7 X_14,3,15,4 X_6,15,7,16 X_16,5,17,6 X_4,17,5,18 X_22,20,11,19 X_18,9,19,10 X_2,11,3,12 X_10,13,1,14 X_8,22,9,21
Gauss code	{1, -9, 3, -6, 5, -4, 2, -11, 8, -10}, {9, -1, 10, -3, 4, -5, 6, -8, 7, -2, 11, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{2}t(1)^{4}-t(2)t(1)^{4}+2t(2)^{3}t(1)^{3}-5t(2)^{2}t(1)^{3}+5t(2)t(1)^{3}-t(1)^{3}+t(2)^{4}t(1)^{2}-5t(2)^{3}t(1)^{2}+9t(2)^{2}t(1)^{2}-5t(2)t(1)^{2}+t(1)^{2}-t(2)^{4}t(1)+5t(2)^{3}t(1)-5t(2)^{2}t(1)+2t(2)t(1)-t(2)^{3}+t(2)^{2}}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+8{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {17}{q^{5/2}}}+{\frac {15}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{3}+2a^{7}z+a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}-2a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z-az^{5}+z^{3}a^{-1}+az$ (db)
Kauffman polynomial	$-a^{6}z^{10}-a^{4}z^{10}-2a^{7}z^{9}-6a^{5}z^{9}-4a^{3}z^{9}-2a^{8}z^{8}-3a^{6}z^{8}-9a^{4}z^{8}-8a^{2}z^{8}-a^{9}z^{7}+4a^{7}z^{7}+11a^{5}z^{7}-4a^{3}z^{7}-10az^{7}+8a^{8}z^{6}+15a^{6}z^{6}+22a^{4}z^{6}+7a^{2}z^{6}-8z^{6}+5a^{9}z^{5}+3a^{7}z^{5}+a^{5}z^{5}+20a^{3}z^{5}+13az^{5}-4z^{5}a^{-1}-10a^{8}z^{4}-14a^{6}z^{4}-9a^{4}z^{4}+4a^{2}z^{4}-z^{4}a^{-2}+8z^{4}-8a^{9}z^{3}-7a^{7}z^{3}-4a^{5}z^{3}-13a^{3}z^{3}-6az^{3}+2z^{3}a^{-1}+4a^{8}z^{2}+5a^{6}z^{2}-3a^{2}z^{2}-2z^{2}+4a^{9}z-2a^{5}z+3a^{3}z+az-a^{6}+a^{7}z^{-1}+a^{5}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

5

1

-4

0

7

3

4

-2

9

6

-3

-4

8

6

2

-6

7

9

2

-8

6

8

-2

-10

3

7

4

-12

2

6

-4

-14

1

4

3

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$