L11n215

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_14,5,15,6 X_9,18,10,19 X_17,22,18,9 X_21,1,22,8 X_20,15,21,16 X_7,16,8,17 X_6,20,7,19 X_4,11,5,12 X_2,13,3,14
Gauss code	{1, -11, 2, -10, 3, -9, -8, 6}, {-4, -1, 10, -2, 11, -3, 7, 8, -5, 4, 9, -7, -6, 5}

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

Polynomial invariants

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

χ

-2

2

-4

3

1

-2

-6

4

1

3

-8

4

3

-1

-10

5

4

1

-12

4

5

1

-14

3

4

-1

-16

1

4

3

-18

1

3

-2

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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