L11n32

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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_9,14,10,15 X₃₈₄₉ X_5,13,6,12 X_13,5,14,22 X_15,18,16,19 X_11,17,12,16 X_17,11,18,10 X_19,2,20,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -9, 6, -7, 4, -8, 9, -10, 8, -11, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+5t(1)t(2)^{3}-t(2)^{3}-t(1)t(2)^{2}+5t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{5/2}+2q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {5}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+a^{5}z^{5}+4a^{5}z^{3}+4a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-9a^{3}z^{3}-10a^{3}z-4a^{3}z^{-1}+2az^{5}+8az^{3}-z^{3}a^{-1}+8az+3az^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-4a^{4}z^{8}-6a^{2}z^{8}-2z^{8}-5a^{5}z^{7}-5a^{3}z^{7}-az^{7}-z^{7}a^{-1}-2a^{6}z^{6}+13a^{4}z^{6}+24a^{2}z^{6}+9z^{6}+18a^{5}z^{5}+32a^{3}z^{5}+19az^{5}+5z^{5}a^{-1}+2a^{6}z^{4}-13a^{4}z^{4}-27a^{2}z^{4}-12z^{4}-4a^{7}z^{3}-25a^{5}z^{3}-44a^{3}z^{3}-31az^{3}-8z^{3}a^{-1}-a^{8}z^{2}-a^{6}z^{2}+7a^{4}z^{2}+13a^{2}z^{2}+6z^{2}+2a^{7}z+14a^{5}z+24a^{3}z+17az+5za^{-1}-a^{6}-2a^{4}-3a^{2}-1-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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

χ

6

1

4

1

-1

2

3

1

2

0

3

1

-2

4

3

1

-4

4

0

-6

3

0

-8

2

4

2

-10

2

3

-1

-12

3

-14

1

-1

Integral Khovanov Homology

(db, data source)

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