L11n207

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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_5,14,6,15 X_17,22,18,9 X_19,5,20,4 X_21,6,22,7 X_7,17,8,16 X_8,9,1,10 X_13,18,14,19 X_15,21,16,20
Gauss code	{1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, -10, 4, -11, 8, -5, 10, -6, 11, -7, 5}

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

Polynomial invariants

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

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

0

1

-1

-2

3

-4

4

2

-2

-6

6

2

4

-8

4

0

-10

6

0

-12

4

5

1

-14

2

5

-3

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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