L11a221

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_16,6,17,5 X_20,13,21,14 X_18,15,19,16 X_14,19,15,20 X_22,10,7,9 X_4,18,5,17 X_10,22,11,21 X₂₇₃₈ X_6,11,1,12
Gauss code	{1, -10, 2, -8, 3, -11}, {10, -1, 7, -9, 11, -2, 4, -6, 5, -3, 8, -5, 6, -4, 9, -7}

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

Polynomial invariants

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

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

1

3

2

6

2

-4

0

8

4

-2

8

7

-1

-4

8

7

1

-6

6

9

3

-8

4

7

-3

-10

2

6

4

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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