L11a240

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-4u^{2}v+3u^{2}-3uv^{4}+7uv^{3}-7uv^{2}+7uv-3u+3v^{4}-4v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+8q^{7/2}-13q^{5/2}+17q^{3/2}-18{\sqrt {q}}+{\frac {18}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+9az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-7a^{3}z+7az-5za^{-1}+za^{-3}+2a^{5}z^{-1}-3a^{3}z^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-3a^{3}z^{9}-8az^{9}-5z^{9}a^{-1}-2a^{4}z^{8}-8a^{2}z^{8}-10z^{8}a^{-2}-16z^{8}-a^{5}z^{7}+8a^{3}z^{7}+14az^{7}-6z^{7}a^{-1}-11z^{7}a^{-3}+6a^{4}z^{6}+34a^{2}z^{6}+12z^{6}a^{-2}-8z^{6}a^{-4}+48z^{6}+5a^{5}z^{5}-4a^{3}z^{5}+5az^{5}+32z^{5}a^{-1}+14z^{5}a^{-3}-4z^{5}a^{-5}-3a^{4}z^{4}-35a^{2}z^{4}-z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-41z^{4}-9a^{5}z^{3}-3a^{3}z^{3}-9az^{3}-23z^{3}a^{-1}-6z^{3}a^{-3}+2z^{3}a^{-5}-4a^{4}z^{2}+8a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+13z^{2}+7a^{5}z+5a^{3}z+3za^{-1}+za^{-3}+3a^{4}+3a^{2}+1-2a^{5}z^{-1}-3a^{3}z^{-1}-az^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

5

1

-4

6

8

3

5

4

9

5

-4

2

9

8

1

0

10

0

-2

5

8

-3

-4

5

10

5

-6

2

5

-3

-8

5

-10

1

2

-1

-12

1

Integral Khovanov Homology

(db, data source)

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