L11a243

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(v^{2}-v+1\right)(uv-u-2v+1)(uv-2u-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {9}{q^{9/2}}}-4q^{7/2}+{\frac {16}{q^{7/2}}}+9q^{5/2}-{\frac {21}{q^{5/2}}}-16q^{3/2}+{\frac {24}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+20{\sqrt {q}}-{\frac {25}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-3a^{3}z+za^{-3}-a^{3}z^{-1}+az^{7}+3az^{5}-2z^{5}a^{-1}+5az^{3}-4z^{3}a^{-1}+5az-3za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-5a^{6}z^{4}+2a^{6}z^{2}+8a^{5}z^{7}-11a^{5}z^{5}+6a^{5}z^{3}-a^{5}z+10a^{4}z^{8}-13a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-2z^{4}a^{-4}+z^{2}a^{-4}-a^{4}+7a^{3}z^{9}+a^{3}z^{7}+4z^{7}a^{-3}-16a^{3}z^{5}-9z^{5}a^{-3}+11a^{3}z^{3}+7z^{3}a^{-3}-2a^{3}z-2za^{-3}+a^{3}z^{-1}+2a^{2}z^{10}+18a^{2}z^{8}+7z^{8}a^{-2}-41a^{2}z^{6}-14z^{6}a^{-2}+26a^{2}z^{4}+7z^{4}a^{-2}-3a^{2}z^{2}-3a^{2}+13az^{9}+6z^{9}a^{-1}-14az^{7}-3z^{7}a^{-1}-10az^{5}-15z^{5}a^{-1}+12az^{3}+15z^{3}a^{-1}-5az-6za^{-1}+3az^{-1}+2a^{-1}z^{-1}+2z^{10}+15z^{8}-39z^{6}+24z^{4}-2z^{2}-3$ (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

1

-5

4

10

3

7

2

11

7

-4

0

14

9

5

-2

11

12

1

-4

10

13

-3

-6

6

11

5

-8

3

10

-7

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$