L11a206

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}-3t(2)^{2}t(1)+5t(2)t(1)-3t(1)-3t(2)+3}{t(1)t(2)}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {7}{q^{11/2}}}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-2za^{9}+z^{5}a^{7}+3z^{3}a^{7}+2za^{7}+z^{5}a^{5}+2z^{3}a^{5}+z^{5}a^{3}+3z^{3}a^{3}+2za^{3}+a^{3}z^{-1}-z^{3}a-3za-az^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-3z^{2}a^{12}-2z^{7}a^{11}+8z^{5}a^{11}-7z^{3}a^{11}+za^{11}-2z^{8}a^{10}+7z^{6}a^{10}-5z^{4}a^{10}+z^{2}a^{10}-2z^{9}a^{9}+9z^{7}a^{9}-15z^{5}a^{9}+12z^{3}a^{9}-2za^{9}-z^{10}a^{8}+4z^{8}a^{8}-7z^{6}a^{8}+6z^{4}a^{8}-2z^{2}a^{8}-3z^{9}a^{7}+15z^{7}a^{7}-29z^{5}a^{7}+19z^{3}a^{7}-3za^{7}-z^{10}a^{6}+5z^{8}a^{6}-12z^{6}a^{6}+11z^{4}a^{6}-6z^{2}a^{6}-z^{9}a^{5}+3z^{7}a^{5}-4z^{5}a^{5}-z^{3}a^{5}+za^{5}-z^{8}a^{4}+2z^{6}a^{4}-2z^{4}a^{4}+z^{2}a^{4}-z^{7}a^{3}+z^{5}a^{3}+3z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{6}a^{2}+2z^{4}a^{2}+z^{2}a^{2}-a^{2}-z^{5}a+4z^{3}a-4za+az^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

-2

3

1

2

-4

2

1

-1

-6

4

2

-8

4

3

-1

-10

3

0

-12

3

4

1

-14

2

3

-1

-16

1

3

2

-18

1

2

-1

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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