L11n278

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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_18,10,19,9 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_10,20,5,19
Gauss code	{1, 4, -3, -10}, {-2, -1, 5, 3, 6, -11}, {-8, 2, -4, -5, 10, 9, -7, -6, 11, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-uv^{2}w+uv-vw^{2}+w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{4}+q^{3}-q^{2}+q+q^{-1}+q^{-2}+q^{-3}+q^{-4}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+6z^{4}-6a^{2}z^{2}-4z^{2}a^{-2}+11z^{2}+2a^{4}-9a^{2}-3a^{-2}+10+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+az^{7}+2z^{7}a^{-1}+z^{7}a^{-3}-a^{2}z^{6}-6z^{6}a^{-2}-7z^{6}-a^{3}z^{5}-8az^{5}-13z^{5}a^{-1}-6z^{5}a^{-3}+a^{4}z^{4}+7a^{2}z^{4}+10z^{4}a^{-2}+16z^{4}+6a^{3}z^{3}+21az^{3}+25z^{3}a^{-1}+10z^{3}a^{-3}-5a^{4}z^{2}-17a^{2}z^{2}-6z^{2}a^{-2}-18z^{2}-11a^{3}z-24az-18za^{-1}-5za^{-3}+7a^{4}+15a^{2}+3a^{-2}+12+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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

5

χ

9

1

-1

7

0

5

1

0

3

1

0

1

-1

1

3

1

-3

1

2

-5

3

1

2

-7

1

2

-9

1

Integral Khovanov Homology

(db, data source)

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