L11a122

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_18,8,19,7 X_22,20,5,19 X_20,9,21,10 X_8,21,9,22 X_16,12,17,11 X_12,16,13,15 X_10,18,11,17 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -8, 11, -2, 8, -7, 9, -3, 4, -5, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{3}-8uv^{2}+10uv-3u-3v^{3}+10v^{2}-8v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+7q^{5/2}-10q^{3/2}+13{\sqrt {q}}-{\frac {15}{\sqrt {q}}}+{\frac {14}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-2a^{5}z^{-1}+3a^{3}z^{3}+z^{3}a^{-3}+3a^{3}z+2a^{3}z^{-1}-az^{5}-z^{5}a^{-1}-z^{3}a^{-1}-az-az^{-1}-za^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+2a^{6}z^{6}-4a^{6}z^{4}+2a^{6}z^{2}+2a^{5}z^{7}+a^{5}z^{5}-9a^{5}z^{3}+8a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}+2a^{4}z^{6}+z^{6}a^{-4}-7a^{4}z^{4}-2z^{4}a^{-4}+4a^{4}z^{2}-a^{4}+2a^{3}z^{9}-a^{3}z^{7}+4z^{7}a^{-3}+6a^{3}z^{5}-11z^{5}a^{-3}-13a^{3}z^{3}+5z^{3}a^{-3}+9a^{3}z-2a^{3}z^{-1}+a^{2}z^{10}+3a^{2}z^{8}+6z^{8}a^{-2}-7a^{2}z^{6}-18z^{6}a^{-2}+8a^{2}z^{4}+13z^{4}a^{-2}-3a^{2}z^{2}-2z^{2}a^{-2}+6az^{9}+4z^{9}a^{-1}-14az^{7}-7z^{7}a^{-1}+14az^{5}-3z^{5}a^{-1}-6az^{3}+6z^{3}a^{-1}+3az-za^{-1}-az^{-1}+z^{10}+7z^{8}-26z^{6}+26z^{4}-7z^{2}$ (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

4

1

-3

4

6

3

2

7

4

-3

0

8

6

2

-2

7

8

1

-4

5

7

-2

-6

3

7

4

-8

2

5

-3

-10

1

4

3

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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