L11a210

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X₆₇₁₈ X_22,15,7,16 X_4,14,5,13 X_16,6,17,5 X_12,18,13,17 X_10,4,11,3 X_18,12,19,11 X_14,21,15,22 X_2,20,3,19
Gauss code	{1, -11, 8, -5, 6, -3}, {3, -1, 2, -8, 9, -7, 5, -10, 4, -6, 7, -9, 11, -2, 10, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+9t(1)t(2)^{3}-4t(2)^{3}+6t(1)^{2}t(2)^{2}-11t(1)t(2)^{2}+6t(2)^{2}-4t(1)^{2}t(2)+9t(1)t(2)-5t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{11/2}+4q^{9/2}-9q^{7/2}+14q^{5/2}-20q^{3/2}+22{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+3az^{3}+3z^{3}a^{-1}-2z^{3}a^{-3}-a^{3}z+az+2za^{-1}-za^{-3}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-10az^{9}-16z^{9}a^{-1}-6z^{9}a^{-3}-15a^{2}z^{8}-5z^{8}a^{-2}-4z^{8}a^{-4}-16z^{8}-14a^{3}z^{7}+8az^{7}+40z^{7}a^{-1}+17z^{7}a^{-3}-z^{7}a^{-5}-9a^{4}z^{6}+24a^{2}z^{6}+37z^{6}a^{-2}+13z^{6}a^{-4}+57z^{6}-4a^{5}z^{5}+19a^{3}z^{5}+17az^{5}-20z^{5}a^{-1}-11z^{5}a^{-3}+3z^{5}a^{-5}-a^{6}z^{4}+7a^{4}z^{4}-11a^{2}z^{4}-38z^{4}a^{-2}-13z^{4}a^{-4}-44z^{4}+a^{5}z^{3}-10a^{3}z^{3}-15az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-3z^{3}a^{-5}-2a^{4}z^{2}+2a^{2}z^{2}+10z^{2}a^{-2}+4z^{2}a^{-4}+10z^{2}+2a^{3}z+3az+za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

3

-3

8

6

1

5

6

8

3

-5

4

12

6

2

11

9

-2

0

12

11

1

-2

9

12

3

-4

6

11

-5

-6

3

9

6

-8

1

6

-5

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

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