L11n350

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_11,21,12,20 X_7,18,8,19 X_9,16,10,17 X_17,8,18,9 X_19,13,20,22 X_13,10,14,11 X_21,5,22,12 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 8, -3, 9}, {-8, -2, 11, 5, -6, 4, -7, 3, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(3)t(2)^{3}+t(2)^{3}-t(1)t(3)^{2}t(2)^{2}+t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)-t(1)t(3)t(2)+t(3)t(2)-t(1)t(3)^{3}+t(1)t(3)^{2}}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+2q^{4}-2q^{3}+2q^{2}-q+2+q^{-1}-q^{-2}+2q^{-3}-q^{-4}+q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{2}a^{4}+a^{4}z^{-2}+2a^{4}-z^{4}a^{2}-4z^{2}a^{2}-2a^{2}z^{-2}-5a^{2}+z^{-2}+2+z^{4}a^{-2}+3z^{2}a^{-2}+2a^{-2}-z^{2}a^{-4}-a^{-4}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+a^{4}z^{8}-7a^{4}z^{6}+2z^{6}a^{-4}+16a^{4}z^{4}-7z^{4}a^{-4}-15a^{4}z^{2}+4z^{2}a^{-4}-a^{4}z^{-2}+6a^{4}-a^{-4}+a^{3}z^{9}-6a^{3}z^{7}+z^{7}a^{-3}+9a^{3}z^{5}-3z^{5}a^{-3}-a^{3}z^{3}-5a^{3}z+za^{-3}+2a^{3}z^{-1}+3a^{2}z^{8}-21a^{2}z^{6}+z^{6}a^{-2}+44a^{2}z^{4}-4z^{4}a^{-2}-36a^{2}z^{2}+2z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+az^{9}-6az^{7}+z^{7}a^{-1}+6az^{5}-7z^{5}a^{-1}+6az^{3}+10z^{3}a^{-1}-8az-3za^{-1}+2az^{-1}+2z^{8}-15z^{6}+31z^{4}-23z^{2}-z^{-2}+9$ (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

4

5

χ

11

1

-1

9

1

7

1

0

5

1

2

1

0

3

1

4

2

1

-1

1

2

4

3

-3

1

0

-5

1

-7

1

2

1

-9

0

-11

1

Integral Khovanov Homology

(db, data source)

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