L11n380

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_17,22,18,19 X_11,20,12,21 X_19,12,20,13 X_21,18,22,5 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {-4, -1, 2, -5, -10, 4, -7, 8, 11, -2, 3, 10, -6, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)^{2}(v+w)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-4q^{-8}+5q^{-7}-4q^{-6}+6q^{-5}-4q^{-4}+4q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+3z^{2}a^{8}+4a^{8}z^{-2}+6a^{8}-2z^{4}a^{6}-7z^{2}a^{6}-5a^{6}z^{-2}-10a^{6}+3z^{2}a^{4}+2a^{4}z^{-2}+4a^{4}+z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{7}a^{11}-5z^{5}a^{11}+8z^{3}a^{11}-5za^{11}+a^{11}z^{-1}+2z^{8}a^{10}-9z^{6}a^{10}+11z^{4}a^{10}-6z^{2}a^{10}-a^{10}z^{-2}+3a^{10}+z^{9}a^{9}-17z^{5}a^{9}+31z^{3}a^{9}-20za^{9}+5a^{9}z^{-1}+5z^{8}a^{8}-23z^{6}a^{8}+31z^{4}a^{8}-21z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+z^{9}a^{7}+z^{7}a^{7}-21z^{5}a^{7}+39z^{3}a^{7}-30za^{7}+9a^{7}z^{-1}+3z^{8}a^{6}-14z^{6}a^{6}+23z^{4}a^{6}-23z^{2}a^{6}-5a^{6}z^{-2}+16a^{6}+2z^{7}a^{5}-9z^{5}a^{5}+17z^{3}a^{5}-15za^{5}+5a^{5}z^{-1}+3z^{4}a^{4}-7z^{2}a^{4}-2a^{4}z^{-2}+6a^{4}+z^{3}a^{3}+z^{2}a^{2}-a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-1

1

-3

2

0

-5

2

1

3

-7

3

0

-9

4

2

-11

2

5

1

2

-13

3

1

-15

1

2

1

-17

1

3

-2

-19

1

-21

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$	$i=-1$
$r=-9$		${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$		${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$