L11a318

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_8,9,1,10 X_20,13,21,14 X_16,8,17,7 X_18,6,19,5 X_6,18,7,17 X_4,20,5,19 X_22,15,9,16 X_14,21,15,22
Gauss code	{1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -11, 10, -6, 8, -7, 9, -5, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}+t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)+t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}-q^{3/2}+{\frac {4}{q^{3/2}}}+{\frac {1}{q^{19/2}}}-{\frac {2}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{7}-4z^{3}a^{7}-3za^{7}+z^{7}a^{5}+5z^{5}a^{5}+6z^{3}a^{5}-a^{5}z^{-1}+z^{7}a^{3}+6z^{5}a^{3}+12z^{3}a^{3}+10za^{3}+3a^{3}z^{-1}-z^{5}a-5z^{3}a-7za-2az^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}+2a^{11}z^{3}+3a^{10}z^{4}-2a^{10}z^{2}+4a^{9}z^{5}-6a^{9}z^{3}+5a^{8}z^{6}-12a^{8}z^{4}+3a^{8}z^{2}+6a^{7}z^{7}-22a^{7}z^{5}+20a^{7}z^{3}-6a^{7}z+5a^{6}z^{8}-21a^{6}z^{6}+23a^{6}z^{4}-8a^{6}z^{2}+a^{6}+3a^{5}z^{9}-12a^{5}z^{7}+8a^{5}z^{5}+5a^{5}z^{3}-a^{5}z^{-1}+a^{4}z^{10}-a^{4}z^{8}-16a^{4}z^{6}+37a^{4}z^{4}-21a^{4}z^{2}+3a^{4}+4a^{3}z^{9}-26a^{3}z^{7}+57a^{3}z^{5}-51a^{3}z^{3}+19a^{3}z-3a^{3}z^{-1}+a^{2}z^{10}-6a^{2}z^{8}+10a^{2}z^{6}-a^{2}z^{4}-7a^{2}z^{2}+3a^{2}+az^{9}-8az^{7}+23az^{5}-28az^{3}+13az-2az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

4

1

2

0

3

1

2

-2

1

-1

-4

4

3

1

-6

3

0

-8

3

2

1

-10

2

3

1

-12

2

3

-1

-14

1

2

1

-16

1

2

-1

-18

1

-20

1

-1

Integral Khovanov Homology

(db, data source)

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