L11n288

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_11,18,12,19 X_9,21,10,20 X_19,5,20,10 X_4,15,1,16 X_17,22,18,11 X_21,16,22,17
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, -4, -5, 9, 11, -10, 6, -8, 7, -11, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(2)^{2}t(3)^{4}+t(1)t(2)t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+t(3)^{2}+t(1)t(2)^{2}t(3)-t(3)-t(2)+1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{-10}+q^{-9}-2q^{-8}+3q^{-7}-2q^{-6}+4q^{-5}-2q^{-4}+3q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-2a^{10}+z^{6}a^{8}+6z^{4}a^{8}+11z^{2}a^{8}+4a^{8}z^{-2}+9a^{8}-z^{8}a^{6}-7z^{6}a^{6}-17z^{4}a^{6}-20z^{2}a^{6}-5a^{6}z^{-2}-14a^{6}+z^{6}a^{4}+6z^{4}a^{4}+11z^{2}a^{4}+2a^{4}z^{-2}+7a^{4}$ (db)
Kauffman polynomial	$a^{13}z+a^{12}z^{2}+2a^{11}z^{3}-4a^{11}z+a^{11}z^{-1}+a^{10}z^{6}-2a^{10}z^{4}-4a^{10}z^{2}-a^{10}z^{-2}+5a^{10}+3a^{9}z^{7}-16a^{9}z^{5}+27a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+3a^{8}z^{8}-17a^{8}z^{6}+32a^{8}z^{4}-32a^{8}z^{2}-4a^{8}z^{-2}+18a^{8}+a^{7}z^{9}-2a^{7}z^{7}-12a^{7}z^{5}+34a^{7}z^{3}-29a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-25a^{6}z^{6}+51a^{6}z^{4}-45a^{6}z^{2}-5a^{6}z^{-2}+21a^{6}+a^{5}z^{9}-5a^{5}z^{7}+4a^{5}z^{5}+9a^{5}z^{3}-13a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-7a^{4}z^{6}+17a^{4}z^{4}-18a^{4}z^{2}-2a^{4}z^{-2}+9a^{4}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

-1

1

-3

0

-5

3

1

2

-7

1

-9

4

2

-11

2

-13

3

2

1

-15

1

2

1

-17

1

2

-1

-19

1

0

-21

1

-1

Integral Khovanov Homology

(db, data source)

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