L11n353

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_5,16,6,17 X₈₄₉₃ X_9,21,10,20 X_19,11,20,10 X_17,22,18,15 X_21,18,22,19 X_15,14,16,5 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {-10, 4, -8, 9, -7, 6, -9, 8}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-2q^{-6}+5q^{-5}-8q^{-4}+q^{3}+11q^{-3}-2q^{2}-10q^{-2}+6q+11q^{-1}-8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}-a^{4}z^{4}+a^{4}+a^{2}z^{6}+3a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{2}+2a^{-2}-2z^{4}-5z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+5a^{2}z^{8}+2z^{8}+3a^{5}z^{7}+5a^{3}z^{7}+4az^{7}+2z^{7}a^{-1}+a^{6}z^{6}-2a^{4}z^{6}-4a^{2}z^{6}+z^{6}a^{-2}-2a^{5}z^{5}-8a^{3}z^{5}-11az^{5}-5z^{5}a^{-1}+4a^{6}z^{4}+3a^{4}z^{4}-11a^{2}z^{4}-4z^{4}a^{-2}-14z^{4}+3a^{7}z^{3}+5a^{5}z^{3}-2a^{3}z^{3}-2az^{3}+2z^{3}a^{-1}-3a^{6}z^{2}-5a^{4}z^{2}+11a^{2}z^{2}+6z^{2}a^{-2}+19z^{2}-2a^{7}z-3a^{5}z+3a^{3}z+7az+3za^{-1}+2a^{6}+2a^{4}-6a^{2}-4a^{-2}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

χ

7

1

5

1

-1

3

5

1

4

1

3

1

-2

-1

8

5

3

-3

6

7

1

-5

5

4

1

-7

3

6

3

-9

2

5

-3

-11

3

-13

2

-2

Integral Khovanov Homology

(db, data source)

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