L11n273

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,7,17,8 X_8,15,5,16 X_11,19,12,18 X_17,9,18,22 X_21,13,22,12 X_13,21,14,20 X_19,15,20,14 X₂₅₃₆ X_4,9,1,10
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 $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(3)^{3}+2t(2)t(3)^{3}-2t(3)^{3}-3t(1)t(3)^{2}-3t(2)t(3)^{2}+t(3)^{2}+3t(1)t(3)-t(1)t(2)t(3)+3t(2)t(3)-2t(1)+2t(1)t(2)-2t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-2q^{3}+3q^{2}-6q+9-8q^{-1}+9q^{-2}-6q^{-3}+6q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-a^{4}z^{-2}-4a^{4}+2z^{4}a^{2}+3z^{2}a^{2}-2a^{2}z^{-2}-a^{2}+2z^{4}+5z^{2}+3z^{-2}+7-2z^{2}a^{-2}-a^{-2}z^{-2}-3a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+4a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-2a^{4}z^{6}-8a^{4}z^{4}+6a^{4}z^{2}+a^{4}z^{-2}-3a^{4}+a^{3}z^{9}+3a^{3}z^{7}-17a^{3}z^{5}+18a^{3}z^{3}+3z^{3}a^{-3}-10a^{3}z-7za^{-3}+2a^{3}z^{-1}+2a^{-3}z^{-1}+6a^{2}z^{8}-19a^{2}z^{6}+z^{6}a^{-2}+19a^{2}z^{4}-8a^{2}z^{2}-z^{2}a^{-2}-2a^{2}z^{-2}-a^{-2}z^{-2}+4a^{2}+a^{-2}+az^{9}+5az^{7}+4z^{7}a^{-1}-29az^{5}-17z^{5}a^{-1}+50az^{3}+35z^{3}a^{-1}-34az-27za^{-1}+10az^{-1}+8a^{-1}z^{-1}+4z^{8}-15z^{6}+23z^{4}-9z^{2}-3z^{-2}+5$ (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

3

χ

7

2

-2

5

1

3

5

2

-3

1

4

1

3

-1

5

6

1

-3

4

1

-5

2

5

3

-7

4

0

-9

1

5

4

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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