L11n38

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_13,19,14,18 X_17,15,18,14 X_15,10,16,11 X_11,22,12,5 X_19,2,20,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, -11, -2, 3, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-4uv^{4}+3uv^{3}-uv^{2}-v^{3}+3v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+a^{9}z^{3}+5a^{9}z+3a^{9}z^{-1}-2a^{7}z^{5}-9a^{7}z^{3}-10a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+5a^{5}z^{5}+8a^{5}z^{3}+7a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{2}-a^{12}+2a^{11}z^{3}-2a^{11}z+a^{11}z^{-1}+a^{10}z^{6}-2a^{10}z^{4}+4a^{10}z^{2}-2a^{10}+3a^{9}z^{7}-12a^{9}z^{5}+18a^{9}z^{3}-11a^{9}z+3a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+11a^{8}z^{4}-4a^{8}z^{2}+a^{7}z^{9}+2a^{7}z^{7}-22a^{7}z^{5}+32a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+5a^{6}z^{8}-21a^{6}z^{6}+24a^{6}z^{4}-11a^{6}z^{2}+2a^{6}+a^{5}z^{9}-15a^{5}z^{5}+24a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{4}-4a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

3

1

2

-6

3

2

-1

-8

3

2

1

-10

1

3

1

-12

4

3

1

-14

1

3

2

-16

1

3

-2

-18

1

-20

1

-1

Integral Khovanov Homology

(db, data source)

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