L11a99

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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_10,5,11,6 X_22,11,5,12 X_4,21,1,22 X_18,16,19,15 X_16,8,17,7 X_8,18,9,17 X_20,14,21,13 X_14,20,15,19
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 10, -11, 7, -8, 9, -7, 11, -10, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4t(1)t(2)^{3}-3t(2)^{3}-10t(1)t(2)^{2}+10t(2)^{2}+10t(1)t(2)-10t(2)-3t(1)+4}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {6}{q^{9/2}}}-4q^{7/2}+{\frac {10}{q^{7/2}}}+8q^{5/2}-{\frac {15}{q^{5/2}}}-11q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+15{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+za^{5}-z^{5}a^{3}-z^{3}a^{3}+2a^{3}z^{-1}-2z^{5}a-4z^{3}a-5za-3az^{-1}-z^{5}a^{-1}+2za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-10az^{9}-6z^{9}a^{-1}-4a^{4}z^{8}-2a^{2}z^{8}-7z^{8}a^{-2}-5z^{8}-4a^{5}z^{7}+6a^{3}z^{7}+30az^{7}+16z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+2a^{4}z^{6}+10a^{2}z^{6}+21z^{6}a^{-2}-z^{6}a^{-4}+27z^{6}-a^{7}z^{5}+5a^{5}z^{5}-9a^{3}z^{5}-38az^{5}-13z^{5}a^{-1}+10z^{5}a^{-3}+6a^{6}z^{4}+5a^{4}z^{4}-14a^{2}z^{4}-16z^{4}a^{-2}+2z^{4}a^{-4}-31z^{4}+2a^{7}z^{3}+9a^{3}z^{3}+20az^{3}+6z^{3}a^{-1}-3z^{3}a^{-3}-3a^{6}z^{2}-3a^{4}z^{2}+7a^{2}z^{2}+4z^{2}a^{-2}+11z^{2}-a^{7}z-4a^{3}z-7az-2za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

5

1

-4

4

6

3

2

9

5

-4

0

9

6

3

-2

8

10

2

-4

7

8

-1

-6

3

8

5

-8

3

7

-4

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$