L11n292

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_11,18,12,19 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_9,11,10,22 X_17,1,18,4 X_19,5,20,10 X_5,12,6,13 X_21,15,22,14 X_13,21,14,20
Gauss code	{1, -4, -3, 7}, {-9, -1, 5, 3, -6, 8}, {-2, 9, -11, 10, 4, -5, -7, 2, -8, 11, -10, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(3)^{4}+t(1)t(2)t(3)^{4}-t(2)^{2}t(3)^{3}+2t(1)t(3)^{3}-2t(1)t(2)t(3)^{3}+t(2)t(3)^{3}+2t(2)^{2}t(3)^{2}-2t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-2t(2)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)+t(2)^{2}-t(2)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{5}-3q^{4}-q^{-4}+5q^{3}+3q^{-3}-6q^{2}-4q^{-2}+8q+7q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-4}+a^{-4}-a^{2}z^{4}-2z^{4}a^{-2}-2a^{2}z^{2}-5z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{-2}+z^{6}+4z^{4}+5z^{2}-2z^{-2}+1$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+3a^{2}z^{8}+5z^{8}a^{-2}+8z^{8}+a^{3}z^{7}-5az^{7}-2z^{7}a^{-1}+4z^{7}a^{-3}-14a^{2}z^{6}-21z^{6}a^{-2}+z^{6}a^{-4}-36z^{6}-4a^{3}z^{5}-6az^{5}-16z^{5}a^{-1}-14z^{5}a^{-3}+19a^{2}z^{4}+30z^{4}a^{-2}+z^{4}a^{-4}+48z^{4}+4a^{3}z^{3}+14az^{3}+24z^{3}a^{-1}+17z^{3}a^{-3}+3z^{3}a^{-5}-9a^{2}z^{2}-20z^{2}a^{-2}-3z^{2}a^{-4}+z^{2}a^{-6}-25z^{2}-a^{3}z-3az-7za^{-1}-7za^{-3}-2za^{-5}+4a^{-2}+2a^{-4}+3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

11

1

9

2

-2

7

3

1

2

5

4

3

-1

3

4

3

1

2

1

4

6

2

-1

3

1

-3

2

5

3

-5

1

2

-1

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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