L11a447

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{3}w^{3}-uv^{3}w^{2}-2uv^{2}w^{3}+4uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+4uvw-uv+uw^{2}-2uw+u-v^{3}w^{3}+2v^{3}w^{2}-v^{3}w+v^{2}w^{3}-4v^{2}w^{2}+4v^{2}w-v^{2}+vw^{2}-4vw+2v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}+13q^{3}-14q^{2}+15q-11+10q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-a^{2}z^{4}-14z^{4}a^{-2}-3a^{2}z^{2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+2z^{6}+9z^{4}+13z^{2}+z^{-2}+8$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-3z^{3}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-8z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+6z^{8}a^{-4}-10z^{6}a^{-4}+8z^{4}a^{-4}-5z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+4z^{9}a^{-3}+a^{3}z^{7}-3z^{7}a^{-3}-4a^{3}z^{5}-10z^{5}a^{-3}+5a^{3}z^{3}+16z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+10z^{8}a^{-2}-12a^{2}z^{6}-41z^{6}a^{-2}+15a^{2}z^{4}+52z^{4}a^{-2}-9a^{2}z^{2}-32z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+3az^{9}+7z^{9}a^{-1}-7az^{7}-17z^{7}a^{-1}-5az^{5}+15az^{3}+18z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+z^{10}+7z^{8}-38z^{6}+52z^{4}-32z^{2}-z^{-2}+11$ (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

2

-2

11

4

1

3

9

6

3

-3

7

3

4

5

7

6

-1

3

8

7

1

6

10

4

-1

4

5

-1

-3

2

6

4

-5

1

4

-3

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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