L11a196

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{2}-5u^{2}v+3u^{2}-5uv^{2}+11uv-5u+3v^{2}-5v+2}{uv}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+5q^{7/2}-9q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+za^{-5}-2a^{3}z^{3}-2z^{3}a^{-3}-2a^{3}z-2za^{-3}-a^{-3}z^{-1}+az^{5}+z^{5}a^{-1}+az^{3}+z^{3}a^{-1}+az+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-6az^{9}-3z^{9}a^{-1}-3a^{4}z^{8}-4a^{2}z^{8}-4z^{8}a^{-2}-5z^{8}-a^{5}z^{7}+10a^{3}z^{7}+18az^{7}+3z^{7}a^{-1}-4z^{7}a^{-3}+13a^{4}z^{6}+26a^{2}z^{6}+3z^{6}a^{-2}-4z^{6}a^{-4}+20z^{6}+4a^{5}z^{5}-6a^{3}z^{5}-14az^{5}-z^{5}a^{-1}-3z^{5}a^{-5}-16a^{4}z^{4}-32a^{2}z^{4}+z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-19z^{4}-4a^{5}z^{3}-2a^{3}z^{3}+4az^{3}+5z^{3}a^{-1}+7z^{3}a^{-3}+4z^{3}a^{-5}+6a^{4}z^{2}+11a^{2}z^{2}+z^{2}a^{-6}+6z^{2}+a^{5}z+a^{3}z-az-4za^{-1}-5za^{-3}-2za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

3

1

-2

6

2

4

6

4

-2

2

7

5

2

0

6

7

1

-2

5

6

-1

-4

3

6

3

-6

2

5

-3

-8

1

3

2

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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