L11a104

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_12,3,13,4 X_20,11,21,12 X_18,6,19,5 X_22,20,5,19 X_10,21,11,22 X_4,17,1,18 X_16,14,17,13 X_14,8,15,7 X_8,16,9,15
Gauss code	{1, -2, 3, -8}, {5, -1, 10, -11, 2, -7, 4, -3, 9, -10, 11, -9, 8, -5, 6, -4, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-6uv^{4}+12uv^{3}-12uv^{2}+4uv+4v^{4}-12v^{3}+12v^{2}-6v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-5q^{7/2}+10q^{5/2}-15q^{3/2}+20{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {19}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-6a^{3}z^{3}+9az^{3}-4z^{3}a^{-1}+z^{3}a^{-3}+2a^{5}z-9a^{3}z+8az-3za^{-1}+2a^{5}z^{-1}-4a^{3}z^{-1}+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-12az^{9}-7z^{9}a^{-1}-7a^{4}z^{8}-12a^{2}z^{8}-9z^{8}a^{-2}-14z^{8}-6a^{5}z^{7}-4a^{3}z^{7}+15az^{7}+8z^{7}a^{-1}-5z^{7}a^{-3}-3a^{6}z^{6}+8a^{4}z^{6}+27a^{2}z^{6}+21z^{6}a^{-2}-z^{6}a^{-4}+38z^{6}-a^{7}z^{5}+10a^{5}z^{5}+24a^{3}z^{5}+11az^{5}+8z^{5}a^{-1}+10z^{5}a^{-3}+4a^{6}z^{4}-3a^{4}z^{4}-14a^{2}z^{4}-11z^{4}a^{-2}+z^{4}a^{-4}-19z^{4}+2a^{7}z^{3}-10a^{5}z^{3}-32a^{3}z^{3}-24az^{3}-7z^{3}a^{-1}-3z^{3}a^{-3}-a^{6}z^{2}-2a^{4}z^{2}-4a^{2}z^{2}-z^{2}a^{-2}-4z^{2}-a^{7}z+7a^{5}z+18a^{3}z+13az+2za^{-1}-za^{-3}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

4

6

1

-5

4

9

4

5

2

11

6

-5

0

12

9

3

-2

11

12

1

-4

8

11

-3

-6

5

11

6

-8

3

8

-5

-10

1

6

5

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$