L11a379

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}v^{2}-u^{4}v-u^{3}v^{4}+4u^{3}v^{3}-6u^{3}v^{2}+4u^{3}v-u^{3}+2u^{2}v^{4}-7u^{2}v^{3}+9u^{2}v^{2}-7u^{2}v+2u^{2}-uv^{4}+4uv^{3}-6uv^{2}+4uv-u-v^{3}+v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-7{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {21}{q^{7/2}}}+{\frac {18}{q^{9/2}}}-{\frac {14}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-a^{7}z+2a^{5}z^{5}+4a^{5}z^{3}+a^{5}z-a^{3}z^{7}-3a^{3}z^{5}-3a^{3}z^{3}-a^{3}z+a^{3}z^{-1}+2az^{5}+5az^{3}-z^{3}a^{-1}+2az-az^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}-4z^{5}a^{9}+z^{3}a^{9}-9z^{6}a^{8}+8z^{4}a^{8}-3z^{2}a^{8}-13z^{7}a^{7}+18z^{5}a^{7}-10z^{3}a^{7}+2za^{7}-12z^{8}a^{6}+15z^{6}a^{6}-2z^{4}a^{6}-7z^{9}a^{5}+21z^{5}a^{5}-11z^{3}a^{5}-2z^{10}a^{4}-13z^{8}a^{4}+39z^{6}a^{4}-25z^{4}a^{4}+6z^{2}a^{4}-11z^{9}a^{3}+25z^{7}a^{3}-11z^{5}a^{3}+3z^{3}a^{3}-5za^{3}+a^{3}z^{-1}-2z^{10}a^{2}-4z^{8}a^{2}+26z^{6}a^{2}-27z^{4}a^{2}+8z^{2}a^{2}-a^{2}-4z^{9}a+11z^{7}a-6z^{5}a-2z^{3}a-za+az^{-1}-3z^{8}+11z^{6}-13z^{4}+5z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-5z^{3}a^{-1}+2za^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

5

1

4

0

7

2

-5

-2

10

5

-4

10

8

-2

-6

11

9

2

-8

8

11

3

-10

6

10

-4

-12

3

8

5

-14

1

6

-5

-16

3

-18

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$