L11n326

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_13,18,14,19 X_9,21,10,20 X_19,5,20,12 X_11,13,12,22 X_21,11,22,10 X_4,17,1,18
Gauss code	{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, -4, -5, 11, 6, -8, 7, -10, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uv^{3}w^{3}+uv^{3}w^{2}+uv^{2}w^{3}-2uv^{2}w^{2}+uv^{2}w+uvw^{2}-v^{2}w-vw^{2}+2vw-v-w+1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-6}+2q^{-5}-2q^{-4}+q^{3}+5q^{-3}-2q^{2}-4q^{-2}+3q+5q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+a^{4}z^{6}-7a^{2}z^{6}+z^{6}+6a^{4}z^{4}-17a^{2}z^{4}+5z^{4}-a^{6}z^{2}+11a^{4}z^{2}-18a^{2}z^{2}+7z^{2}-2a^{6}+7a^{4}-9a^{2}+4+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+4a^{2}z^{8}+2z^{8}+a^{5}z^{7}-3a^{3}z^{7}-2az^{7}+2z^{7}a^{-1}-12a^{4}z^{6}-21a^{2}z^{6}+z^{6}a^{-2}-8z^{6}-5a^{5}z^{5}-3a^{3}z^{5}-6az^{5}-8z^{5}a^{-1}+2a^{6}z^{4}+28a^{4}z^{4}+37a^{2}z^{4}-4z^{4}a^{-2}+7z^{4}+a^{7}z^{3}+11a^{5}z^{3}+17a^{3}z^{3}+13az^{3}+6z^{3}a^{-1}-5a^{6}z^{2}-27a^{4}z^{2}-28a^{2}z^{2}+3z^{2}a^{-2}-3z^{2}-2a^{7}z-7a^{5}z-12a^{3}z-8az-za^{-1}+3a^{6}+11a^{4}+11a^{2}-a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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

2

1

2

1

0

-1

1

5

2

-3

3

4

1

-5

3

2

1

-7

1

3

-9

2

0

-11

1

-13

1

-1

Integral Khovanov Homology

(db, data source)

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