L11n272

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,17,8,16 X_15,5,16,8 X_18,11,19,12 X_22,17,9,18 X_12,21,13,22 X_20,13,21,14 X_14,19,15,20 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -7, 8, -9, -4, 3, 6, -5, 9, -8, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(3)^{3}-2t(1)t(3)^{2}-2t(2)t(3)^{2}-t(3)^{2}+2t(1)t(3)+t(1)t(2)t(3)+2t(2)t(3)-2t(1)t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-q^{2}+2q+1+q^{-1}+q^{-2}-q^{-3}+2q^{-4}-2q^{-5}+q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+a^{4}z^{4}+4a^{4}z^{2}+3a^{4}z^{-2}+6a^{4}-a^{2}z^{2}-2a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}-3a^{2}+2a^{-2}-z^{4}-4z^{2}-z^{-2}-3$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+11a^{7}z^{3}-7a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+6a^{6}z^{4}-2a^{6}z^{2}-a^{6}z^{-2}+a^{6}+3a^{5}z^{7}-18a^{5}z^{5}+33a^{5}z^{3}-27a^{5}z+8a^{5}z^{-1}+a^{4}z^{8}-5a^{4}z^{6}+5a^{4}z^{4}-4a^{4}z^{2}-3a^{4}z^{-2}+5a^{4}+3a^{3}z^{7}-21a^{3}z^{5}+42a^{3}z^{3}-34a^{3}z+10a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+z^{6}a^{-2}+7a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+6z^{2}a^{-2}-2a^{2}z^{-2}+a^{-2}z^{-2}+4a^{2}-4a^{-2}+2az^{7}+z^{7}a^{-1}-13az^{5}-4z^{5}a^{-1}+20az^{3}-10az+4za^{-1}+2az^{-1}-2a^{-1}z^{-1}+z^{8}-5z^{6}+3z^{4}+5z^{2}+z^{-2}-3$ (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

3

4

χ

7

1

5

1

0

3

1

3

1

3

-1

2

5

1

2

-3

1

2

-5

1

3

2

0

-7

2

1

-9

1

0

-11

1

2

-1

-13

0

-15

1

-1

Integral Khovanov Homology

(db, data source)

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