L11a380

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}v^{2}-u^{4}v-u^{3}v^{4}+4u^{3}v^{3}-5u^{3}v^{2}+3u^{3}v-u^{3}+2u^{2}v^{4}-6u^{2}v^{3}+7u^{2}v^{2}-6u^{2}v+2u^{2}-uv^{4}+3uv^{3}-5uv^{2}+4uv-u-v^{3}+v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-7{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {17}{q^{5/2}}}-{\frac {18}{q^{7/2}}}+{\frac {15}{q^{9/2}}}-{\frac {12}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+2z^{5}a^{5}+5z^{3}a^{5}+2za^{5}-z^{7}a^{3}-3z^{5}a^{3}-2z^{3}a^{3}+a^{3}z^{-1}+2z^{5}a+5z^{3}a+za-az^{-1}-z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-2a^{9}z^{3}+6a^{8}z^{6}-6a^{8}z^{4}+3a^{8}z^{2}+9a^{7}z^{7}-15a^{7}z^{5}+14a^{7}z^{3}-5a^{7}z+9a^{6}z^{8}-15a^{6}z^{6}+9a^{6}z^{4}-a^{6}z^{2}+6a^{5}z^{9}-5a^{5}z^{7}-11a^{5}z^{5}+14a^{5}z^{3}-5a^{5}z+2a^{4}z^{10}+8a^{4}z^{8}-31a^{4}z^{6}+25a^{4}z^{4}-7a^{4}z^{2}+10a^{3}z^{9}-27a^{3}z^{7}+19a^{3}z^{5}-8a^{3}z^{3}+4a^{3}z-a^{3}z^{-1}+2a^{2}z^{10}+2a^{2}z^{8}-21a^{2}z^{6}+21a^{2}z^{4}-6a^{2}z^{2}+a^{2}+4az^{9}-12az^{7}+z^{7}a^{-1}+8az^{5}-4z^{5}a^{-1}-az^{3}+5z^{3}a^{-1}+2az-az^{-1}-2za^{-1}+3z^{8}-11z^{6}+12z^{4}-4z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

5

1

4

0

6

2

-4

-2

9

5

4

-4

9

7

-2

-6

9

8

1

-8

7

10

3

-10

5

8

-3

-12

2

7

5

-14

1

5

-4

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$