L11n379

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₃₈₄₉ X_22,18,19,17 X_20,12,21,11 X_12,20,13,19 X_18,22,5,21 X_9,16,10,17 X_13,2,14,3
Gauss code	{1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, -10, 4, 7, -8, -11, -2, 3, 10, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)(t(3)+1)\left(t(1)t(3)^{3}+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{2}+1+q^{-1}+q^{-2}+q^{-3}-q^{-4}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-a^{6}+a^{4}z^{4}+6a^{4}z^{2}+4a^{4}z^{-2}+8a^{4}-a^{2}z^{6}-7a^{2}z^{4}-15a^{2}z^{2}-5a^{2}z^{-2}-13a^{2}+z^{4}+5z^{2}+2z^{-2}+6$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}+a^{3}z^{7}+az^{7}-a^{4}z^{6}-9a^{2}z^{6}-8z^{6}-a^{5}z^{5}-9a^{3}z^{5}-8az^{5}+7a^{4}z^{4}+28a^{2}z^{4}+21z^{4}+6a^{5}z^{3}+26a^{3}z^{3}+20az^{3}-15a^{4}z^{2}-37a^{2}z^{2}-22z^{2}-9a^{5}z-27a^{3}z-18az+a^{6}+12a^{4}+21a^{2}+11+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

5

1

3

1

-1

2

-3

3

1

2

-5

2

-7

1

0

-9

1

2

-1

-11

1

0

Integral Khovanov Homology

(db, data source)

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