L11a248

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+6uv-2u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-10q^{9/2}+15q^{7/2}-21q^{5/2}+23q^{3/2}-23{\sqrt {q}}+{\frac {19}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+2z^{5}a^{-3}-a^{3}z^{3}+4z^{3}a^{-3}-a^{3}z+3za^{-3}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-9z^{3}a^{-1}+5az-7za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-4az^{9}-9z^{9}a^{-1}-5z^{9}a^{-3}-6a^{2}z^{8}-21z^{8}a^{-2}-10z^{8}a^{-4}-17z^{8}-4a^{3}z^{7}-5az^{7}-5z^{7}a^{-1}-14z^{7}a^{-3}-10z^{7}a^{-5}-a^{4}z^{6}+12a^{2}z^{6}+41z^{6}a^{-2}+9z^{6}a^{-4}-5z^{6}a^{-6}+40z^{6}+10a^{3}z^{5}+30az^{5}+47z^{5}a^{-1}+43z^{5}a^{-3}+15z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-6a^{2}z^{4}-19z^{4}a^{-2}+4z^{4}a^{-4}+5z^{4}a^{-6}-26z^{4}-8a^{3}z^{3}-30az^{3}-46z^{3}a^{-1}-30z^{3}a^{-3}-6z^{3}a^{-5}-a^{4}z^{2}-3z^{2}a^{-4}+4z^{2}+2a^{3}z+10az+14za^{-1}+6za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

4

-4

10

6

1

5

8

9

4

-5

6

12

6

4

11

9

-2

2

12

0

9

13

4

-2

5

10

-5

-4

3

9

6

-6

1

5

-4

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$