L11a409

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_8,18,9,17 X_14,8,15,7 X_18,10,19,9 X_10,12,5,11 X_22,19,11,20 X_20,15,21,16 X_16,21,17,22 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -3, 5, -6}, {6, -2, 11, -4, 8, -9, 3, -5, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}-uvw^{4}+4uvw^{3}-5uvw^{2}+2uvw-2uw^{3}+4uw^{2}-3uw+u-v^{2}w^{4}+3v^{2}w^{3}-4v^{2}w^{2}+2v^{2}w-2vw^{3}+5vw^{2}-4vw+v-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-11q^{4}-q^{-4}+16q^{3}+3q^{-3}-16q^{2}-7q^{-2}+18q+11q^{-1}-14$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-6z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-15z^{4}a^{-2}+4z^{4}a^{-4}+9z^{4}-3a^{2}z^{2}-19z^{2}a^{-2}+6z^{2}a^{-4}+15z^{2}-3a^{2}-13a^{-2}+4a^{-4}+12-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-7z^{4}a^{-6}+5z^{2}a^{-6}-a^{-6}+8z^{7}a^{-5}-10z^{5}a^{-5}+5z^{3}a^{-5}+9z^{8}a^{-4}-18z^{6}a^{-4}+21z^{4}a^{-4}-17z^{2}a^{-4}-2a^{-4}z^{-2}+8a^{-4}+5z^{9}a^{-3}+a^{3}z^{7}+z^{7}a^{-3}-4a^{3}z^{5}-26z^{5}a^{-3}+6a^{3}z^{3}+33z^{3}a^{-3}-4a^{3}z-18za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+16z^{8}a^{-2}-11a^{2}z^{6}-58z^{6}a^{-2}+13a^{2}z^{4}+72z^{4}a^{-2}-7a^{2}z^{2}-50z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+3a^{2}+20a^{-2}+3az^{9}+8z^{9}a^{-1}-4az^{7}-12z^{7}a^{-1}-15az^{5}-24z^{5}a^{-1}+30az^{3}+50z^{3}a^{-1}-19az-33za^{-1}+5az^{-1}+9a^{-1}z^{-1}+z^{10}+10z^{8}-45z^{6}+56z^{4}-34z^{2}-4z^{-2}+15$ (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

15

1

13

3

1

-2

11

4

9

7

3

-4

7

9

4

5

8

0

3

10

8

2

1

6

10

4

-1

5

8

-3

2

6

4

-5

1

5

-4

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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