L11n127

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_4,15,5,16 X_5,13,6,12 X_11,16,12,17 X_17,6,18,1 X_14,20,15,19 X_20,14,21,13 X_18,21,19,22
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -11, 9, -10, 11, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}-2u^{2}v+u^{2}-uv^{2}+uv-u+v^{2}-2v+1}{uv}}$ (db)
Jones polynomial	$-q^{5/2}+2q^{3/2}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+5az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-4a^{2}z^{8}-2z^{8}-a^{5}z^{7}+2a^{3}z^{7}+2az^{7}-z^{7}a^{-1}+10a^{4}z^{6}+20a^{2}z^{6}+10z^{6}+5a^{5}z^{5}+9a^{3}z^{5}+9az^{5}+5z^{5}a^{-1}-13a^{4}z^{4}-26a^{2}z^{4}-13z^{4}-7a^{5}z^{3}-20a^{3}z^{3}-20az^{3}-7z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+5z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

6

1

4

1

-1

2

1

0

2

0

-2

2

1

-4

2

3

1

-6

1

2

1

0

-8

1

2

1

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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