L11n246

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X₃₈₄₉ X_5,14,6,15 X_7,18,8,19 X_9,21,10,20 X_10,11,1,12 X_13,6,14,7 X_17,4,18,5 X_15,11,16,22 X_19,3,20,2 X_21,17,22,16
Gauss code	{1, 10, -2, 8, -3, 7, -4, 2, -5, -6}, {6, -1, -7, 3, -9, 11, -8, 4, -10, 5, -11, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u+v)\left(u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+5uv-2u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{3/2}+4{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {10}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}-7a^{3}z-a^{3}z^{-1}+3az^{3}+3az-za^{-1}$ (db)
Kauffman polynomial	$-2a^{5}z^{9}-2a^{3}z^{9}-5a^{6}z^{8}-9a^{4}z^{8}-4a^{2}z^{8}-4a^{7}z^{7}-4a^{5}z^{7}-2a^{3}z^{7}-2az^{7}-a^{8}z^{6}+12a^{6}z^{6}+23a^{4}z^{6}+10a^{2}z^{6}+11a^{7}z^{5}+24a^{5}z^{5}+15a^{3}z^{5}+2az^{5}+2a^{8}z^{4}-3a^{6}z^{4}-15a^{4}z^{4}-14a^{2}z^{4}-4z^{4}-7a^{7}z^{3}-20a^{5}z^{3}-19a^{3}z^{3}-7az^{3}-z^{3}a^{-1}-a^{8}z^{2}-a^{6}z^{2}+2a^{4}z^{2}+4a^{2}z^{2}+2z^{2}+a^{7}z+7a^{5}z+8a^{3}z+3az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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

4

1

2

3

-3

0

5

1

4

-2

6

4

-2

-4

6

4

2

-6

5

6

1

-8

5

6

-1

-10

3

6

3

-12

1

4

-3

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

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