L11n286

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_15,1,16,4 X_21,18,22,19 X_9,21,10,20 X_19,5,20,10 X_11,16,12,17 X_17,22,18,11
Gauss code	{1, -4, -3, 6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, 4, -5, -6, 10, -11, 7, -9, 8, -7, 11}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uvw^{2}-3uvw-2uw^{2}+2uw-2v^{2}w+2v^{2}+3vw-2v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{-7}+2q^{-6}-3q^{-5}+5q^{-4}-5q^{-3}+2q^{2}+6q^{-2}-2q-5q^{-1}+5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+z^{4}a^{4}+2z^{2}a^{4}+2a^{4}+z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}+a^{2}-3z^{2}-2z^{-2}-4+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{9}+a^{3}z^{9}+2a^{6}z^{8}+4a^{4}z^{8}+2a^{2}z^{8}+a^{7}z^{7}-2a^{5}z^{7}-a^{3}z^{7}+2az^{7}-10a^{6}z^{6}-17a^{4}z^{6}-5a^{2}z^{6}+2z^{6}-5a^{7}z^{5}-5a^{5}z^{5}-4a^{3}z^{5}-3az^{5}+z^{5}a^{-1}+15a^{6}z^{4}+23a^{4}z^{4}+3a^{2}z^{4}-5z^{4}+7a^{7}z^{3}+10a^{5}z^{3}+2a^{3}z^{3}-2az^{3}-z^{3}a^{-1}-8a^{6}z^{2}-16a^{4}z^{2}-a^{2}z^{2}+3z^{2}a^{-2}+10z^{2}-2a^{7}z-4a^{5}z+6az+4za^{-1}+2a^{6}+4a^{4}-2a^{2}-4a^{-2}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

5

2

3

1

0

1

4

1

3

-1

3

0

-3

3

2

1

-5

3

4

1

-7

2

0

-9

1

3

2

-11

1

2

-1

-13

1

-15

1

-1

Integral Khovanov Homology

(db, data source)

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