L11a211

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,5,17,6 X_14,7,15,8 X_22,15,7,16 X_20,18,21,17 X_18,12,19,11 X_12,20,13,19 X_4,13,5,14 X_6,21,1,22
Gauss code	{1, -2, 3, -10, 4, -11}, {5, -1, 2, -3, 8, -9, 10, -5, 6, -4, 7, -8, 9, -7, 11, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-2t(2)^{3}+6t(1)^{2}t(2)^{2}-11t(1)t(2)^{2}+6t(2)^{2}-2t(1)^{2}t(2)+8t(1)t(2)-5t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {17}{q^{5/2}}}-{\frac {20}{q^{7/2}}}+{\frac {19}{q^{9/2}}}-{\frac {17}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {7}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+3za^{7}+a^{7}z^{-1}-z^{7}a^{5}-4z^{5}a^{5}-7z^{3}a^{5}-6za^{5}-a^{5}z^{-1}-z^{7}a^{3}-3z^{5}a^{3}-2z^{3}a^{3}+z^{5}a+2z^{3}a$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+5z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+7z^{5}a^{9}-3z^{3}a^{9}+za^{9}-6z^{8}a^{8}+8z^{6}a^{8}-5z^{4}a^{8}+2z^{2}a^{8}-5z^{9}a^{7}+5z^{7}a^{7}-4z^{5}a^{7}+5z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-2z^{10}a^{6}-7z^{8}a^{6}+21z^{6}a^{6}-22z^{4}a^{6}+9z^{2}a^{6}-a^{6}-11z^{9}a^{5}+26z^{7}a^{5}-27z^{5}a^{5}+16z^{3}a^{5}-6za^{5}+a^{5}z^{-1}-2z^{10}a^{4}-8z^{8}a^{4}+30z^{6}a^{4}-28z^{4}a^{4}+8z^{2}a^{4}-6z^{9}a^{3}+12z^{7}a^{3}-5z^{5}a^{3}+z^{3}a^{3}-7z^{8}a^{2}+19z^{6}a^{2}-14z^{4}a^{2}+3z^{2}a^{2}-4z^{7}a+10z^{5}a-5z^{3}a-z^{6}+2z^{4}$ (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

4

1

-1

2

3

0

5

1

-4

-2

8

3

5

-4

10

6

-4

-6

10

7

3

-8

9

10

1

-10

8

10

-2

-12

4

9

5

-14

3

8

-5

-16

1

5

4

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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