L11n172

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_3,10,4,11 X_12,7,13,8 X_22,15,7,16 X_14,6,15,5 X_6,14,1,13 X_16,21,17,22 X_9,18,10,19 X_20,11,21,12 X_4,18,5,17 X_19,3,20,2
Gauss code	{1, 11, -2, -10, 5, -6}, {3, -1, -8, 2, 9, -3, 6, -5, 4, -7, 10, 8, -11, -9, 7, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{2}-3u^{2}v+2u^{2}-5uv^{2}+9uv-5u+2v^{2}-3v+1}{uv}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {10}{q^{7/2}}}-{\frac {11}{q^{5/2}}}-2q^{3/2}+{\frac {9}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+4{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+za^{5}-a^{5}z^{-1}-z^{5}a^{3}-z^{3}a^{3}+2a^{3}z^{-1}+3z^{3}a+3za-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-3a^{8}z^{4}+2a^{8}z^{2}+3a^{7}z^{7}-9a^{7}z^{5}+6a^{7}z^{3}-a^{7}z+4a^{6}z^{8}-12a^{6}z^{6}+10a^{6}z^{4}-6a^{6}z^{2}+2a^{6}+2a^{5}z^{9}-a^{5}z^{7}-8a^{5}z^{5}+5a^{5}z^{3}-a^{5}z^{-1}+8a^{4}z^{8}-25a^{4}z^{6}+29a^{4}z^{4}-18a^{4}z^{2}+5a^{4}+2a^{3}z^{9}-a^{3}z^{7}-4a^{3}z^{5}+4a^{3}z^{3}+2a^{3}z-2a^{3}z^{-1}+4a^{2}z^{8}-11a^{2}z^{6}+18a^{2}z^{4}-11a^{2}z^{2}+3a^{2}+3az^{7}-5az^{5}+8az^{3}+3z^{3}a^{-1}-3az-4za^{-1}+a^{-1}z^{-1}+z^{6}+2z^{4}-z^{2}-1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

2

3

1

-2

0

5

1

4

-2

5

4

-1

-4

6

4

2

-6

4

5

1

-8

4

6

-2

-10

2

4

2

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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