L11a247

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)^{3}(v-1)^{3}}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-8q^{9/2}+13q^{7/2}-18q^{5/2}+20q^{3/2}-21{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-8z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+4az-6za^{-1}+4za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+2z^{2}a^{-6}+7z^{7}a^{-5}-11z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}+7z^{8}a^{-4}+a^{4}z^{6}-6z^{6}a^{-4}-2a^{4}z^{4}-3z^{4}a^{-4}+a^{4}z^{2}+3z^{2}a^{-4}+4z^{9}a^{-3}+4a^{3}z^{7}+8z^{7}a^{-3}-10a^{3}z^{5}-28z^{5}a^{-3}+7a^{3}z^{3}+25z^{3}a^{-3}-2a^{3}z-8za^{-3}+z^{10}a^{-2}+6a^{2}z^{8}+15z^{8}a^{-2}-13a^{2}z^{6}-32z^{6}a^{-2}+6a^{2}z^{4}+20z^{4}a^{-2}-2z^{2}a^{-2}+4az^{9}+8z^{9}a^{-1}+2az^{7}-z^{7}a^{-1}-24az^{5}-30z^{5}a^{-1}+24az^{3}+35z^{3}a^{-1}-8az-12za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{10}+14z^{8}-36z^{6}+25z^{4}-4z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

3

-5

6

10

5

4

10

8

-2

2

11

10

1

0

8

12

4

-2

5

9

-4

3

8

5

-6

1

5

-4

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$