L11a327

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_18,10,19,9 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_22,18,9,17 X_16,22,17,21 X_4,20,5,19 X_20,6,21,5
Gauss code	{1, -2, 3, -10, 11, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -9, 8, -4, 10, -11, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-u^{3}v^{5}+u^{3}v^{4}-u^{3}v^{3}+u^{3}v^{2}+u^{2}v^{5}-4u^{2}v^{4}+5u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+2uv^{4}-5uv^{3}+5uv^{2}-4uv+u+v^{3}-v^{2}+v-1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+5q^{5/2}-9q^{3/2}+11{\sqrt {q}}-{\frac {14}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+6a^{3}z^{5}+13a^{3}z^{3}+10a^{3}z+2a^{3}z^{-1}-az^{9}-8az^{7}+z^{7}a^{-1}-25az^{5}+6z^{5}a^{-1}-36az^{3}+13z^{3}a^{-1}-22az+10za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+2a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-5a^{5}z^{5}+2a^{5}z^{3}-a^{5}z+4a^{4}z^{8}-10a^{4}z^{6}+z^{6}a^{-4}+14a^{4}z^{4}-4z^{4}a^{-4}-7a^{4}z^{2}+4z^{2}a^{-4}+3a^{3}z^{9}-6a^{3}z^{7}+2z^{7}a^{-3}+10a^{3}z^{5}-6z^{5}a^{-3}-10a^{3}z^{3}+4z^{3}a^{-3}+9a^{3}z-2a^{3}z^{-1}+a^{2}z^{10}+4a^{2}z^{8}+3z^{8}a^{-2}-17a^{2}z^{6}-9z^{6}a^{-2}+30a^{2}z^{4}+10z^{4}a^{-2}-18a^{2}z^{2}-7z^{2}a^{-2}+3a^{2}+a^{-2}+6az^{9}+3z^{9}a^{-1}-21az^{7}-10z^{7}a^{-1}+43az^{5}+21z^{5}a^{-1}-47az^{3}-28z^{3}a^{-1}+24az+12za^{-1}-3az^{-1}-a^{-1}z^{-1}+z^{10}+3z^{8}-15z^{6}+26z^{4}-21z^{2}+3$ (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

1

6

4

1

-3

4

5

1

4

2

6

4

-2

0

8

5

3

-2

6

7

1

-4

6

7

-1

-6

3

6

3

-8

2

6

-4

-10

1

4

3

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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