L11n270

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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_13,21,14,20 X_19,13,20,12 X_21,15,22,14 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, -5, 8, -7, 9, 4, -3, -6, 5, -8, 7, -9, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{5}+t(2)t(3)^{5}-t(3)^{5}-t(1)t(3)^{4}-t(2)t(3)^{4}+t(1)t(3)^{3}+t(2)t(3)^{3}-t(1)t(3)^{2}-t(2)t(3)^{2}+t(1)t(3)+t(2)t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{-5}-2q^{4}-q^{-4}+2q^{3}+4q^{-3}-3q^{2}-2q^{-2}+4q+5q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$2z^{6}-3a^{2}z^{4}-2z^{4}a^{-2}+12z^{4}+a^{4}z^{2}-14a^{2}z^{2}-8z^{2}a^{-2}+24z^{2}+4a^{4}-18a^{2}-7a^{-2}+21+3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}+7z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+5a^{2}z^{8}+4z^{8}-4a^{3}z^{7}+4z^{7}a^{-1}-7a^{4}z^{6}-30a^{2}z^{6}+2z^{6}a^{-2}-21z^{6}-2a^{3}z^{5}-21az^{5}-17z^{5}a^{-1}+2z^{5}a^{-3}+18a^{4}z^{4}+59a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}+38z^{4}+22a^{3}z^{3}+50az^{3}+24z^{3}a^{-1}-4z^{3}a^{-3}-22a^{4}z^{2}-51a^{2}z^{2}-8z^{2}a^{-2}-37z^{2}-24a^{3}z-45az-21za^{-1}+3za^{-3}+3za^{-5}+13a^{4}+28a^{2}+7a^{-2}+a^{-4}+22+8a^{3}z^{-1}+15az^{-1}+7a^{-1}z^{-1}-a^{-3}z^{-1}-a^{-5}z^{-1}-3a^{4}z^{-2}-8a^{2}z^{-2}-2a^{-2}z^{-2}-7z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

9

2

-2

7

1

0

5

3

2

-1

3

1

4

0

-1

1

-3

1

4

3

-5

3

1

2

-7

1

4

3

-9

0

-11

1

Integral Khovanov Homology

(db, data source)

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