L11a355

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)+t(2)-1)(t(2)t(1)-t(1)-t(2))\left(t(1)^{2}t(2)^{2}+t(1)t(2)+1\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {7}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-4z^{3}a^{9}-3za^{9}+z^{7}a^{7}+5z^{5}a^{7}+8z^{3}a^{7}+6za^{7}+a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+2z^{3}a^{5}-3za^{5}-a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-3za^{3}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-2z^{5}a^{13}+4z^{3}a^{13}-2za^{13}-2z^{6}a^{12}+2z^{4}a^{12}-2z^{7}a^{11}+2z^{5}a^{11}-3za^{11}-2z^{8}a^{10}+4z^{6}a^{10}-5z^{4}a^{10}+z^{2}a^{10}-2z^{9}a^{9}+7z^{7}a^{9}-13z^{5}a^{9}+10z^{3}a^{9}-2za^{9}-z^{10}a^{8}+2z^{8}a^{8}-3z^{4}a^{8}+4z^{2}a^{8}-4z^{9}a^{7}+18z^{7}a^{7}-30z^{5}a^{7}+26z^{3}a^{7}-9za^{7}+a^{7}z^{-1}-z^{10}a^{6}+2z^{8}a^{6}+3z^{6}a^{6}-5z^{4}a^{6}+3z^{2}a^{6}-a^{6}-2z^{9}a^{5}+8z^{7}a^{5}-8z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+a^{5}z^{-1}-2z^{8}a^{4}+9z^{6}a^{4}-10z^{4}a^{4}+2z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+3za^{3}$ (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

1

2

χ

0

1

-2

1

-1

-4

3

1

2

-6

3

2

-1

-8

4

2

-10

4

3

-1

-12

4

0

-14

3

4

1

-16

3

4

-1

-18

1

4

3

-20

1

2

-1

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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