L11a254

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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_14,5,15,6 X_4,13,5,14 X_20,16,21,15 X_16,8,17,7 X_6,21,7,22 X_18,9,19,10 X_22,17,9,18 X_8,20,1,19
Gauss code	{1, -2, 3, -5, 4, -8, 7, -11}, {9, -1, 2, -3, 5, -4, 6, -7, 10, -9, 11, -6, 8, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{5}-2u^{3}v^{4}+2u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+3u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-3uv^{2}+3uv-u-v^{3}+2v^{2}-2v+1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {10}{q^{9/2}}}+{\frac {8}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {3}{q^{19/2}}}-{\frac {5}{q^{17/2}}}+{\frac {7}{q^{15/2}}}-{\frac {9}{q^{13/2}}}+{\frac {10}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{5}z^{9}+a^{7}z^{7}-7a^{5}z^{7}+a^{3}z^{7}+5a^{7}z^{5}-17a^{5}z^{5}+5a^{3}z^{5}+7a^{7}z^{3}-17a^{5}z^{3}+6a^{3}z^{3}+3a^{7}z-5a^{5}z+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{13}z^{3}+3a^{12}z^{4}-a^{12}z^{2}+5a^{11}z^{5}-4a^{11}z^{3}+a^{11}z+6a^{10}z^{6}-7a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-9a^{9}z^{5}-a^{9}z^{3}+a^{9}z+6a^{8}z^{8}-15a^{8}z^{6}+8a^{8}z^{4}-3a^{8}z^{2}+5a^{7}z^{9}-17a^{7}z^{7}+17a^{7}z^{5}-11a^{7}z^{3}+4a^{7}z+2a^{6}z^{10}-2a^{6}z^{8}-17a^{6}z^{6}+27a^{6}z^{4}-9a^{6}z^{2}+8a^{5}z^{9}-40a^{5}z^{7}+61a^{5}z^{5}-33a^{5}z^{3}+5a^{5}z+a^{5}z^{-1}+2a^{4}z^{10}-7a^{4}z^{8}-a^{4}z^{6}+16a^{4}z^{4}-7a^{4}z^{2}-a^{4}+3a^{3}z^{9}-17a^{3}z^{7}+30a^{3}z^{5}-18a^{3}z^{3}+a^{3}z+a^{3}z^{-1}+a^{2}z^{8}-5a^{2}z^{6}+7a^{2}z^{4}-3a^{2}z^{2}$ (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

2

3

χ

2

1

-1

0

2

-2

2

1

-1

-4

5

2

3

-6

4

3

-1

-8

6

4

2

-10

4

0

-12

5

6

-1

-14

3

5

2

-16

2

4

-2

-18

1

3

2

-20

2

-2

-22

1

Integral Khovanov Homology

(db, data source)

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