L11n358

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,19,12,18 X_7,16,8,17 X_15,8,16,9 X_17,15,18,22 X_21,13,22,12 X_13,21,14,20 X_19,5,20,14 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -9, 8, -7, 6}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(t(2)t(3)^{3}-t(3)^{3}+t(2)^{2}t(3)^{2}-2t(2)t(3)^{2}+t(1)t(3)-2t(1)t(2)t(3)-t(1)t(2)^{2}+t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{3}+3q^{2}-4q+7-6q^{-1}+7q^{-2}-5q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-2a^{4}z^{2}+a^{4}z^{-2}-a^{4}+a^{2}z^{4}-2a^{2}z^{-2}-z^{2}a^{-2}-2a^{2}+z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+5a^{2}z^{8}+3z^{8}+2a^{5}z^{7}+2z^{7}a^{-1}+a^{6}z^{6}-6a^{4}z^{6}-22a^{2}z^{6}-15z^{6}-7a^{5}z^{5}-10a^{3}z^{5}-11az^{5}-8z^{5}a^{-1}-4a^{6}z^{4}+4a^{4}z^{4}+37a^{2}z^{4}+3z^{4}a^{-2}+32z^{4}+5a^{5}z^{3}+15a^{3}z^{3}+24az^{3}+15z^{3}a^{-1}+z^{3}a^{-3}+4a^{6}z^{2}-5a^{4}z^{2}-30a^{2}z^{2}-5z^{2}a^{-2}-26z^{2}-a^{5}z-10a^{3}z-14az-7za^{-1}-2za^{-3}-a^{6}+4a^{4}+12a^{2}+2a^{-2}+10+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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

7

1

-1

5

2

3

2

-1

1

4

1

3

-1

3

4

1

-3

4

3

1

-5

2

4

2

-7

2

3

-1

-9

1

3

2

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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