L11n66

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_7,14,8,15 X_15,22,16,5 X_9,17,10,16 X_21,9,22,8 X_17,21,18,20 X_13,18,14,19 X_19,12,20,13 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 6, -5, 2, -11, 9, -8, 3, -4, 5, -7, 8, -9, 7, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{5}-5uv^{4}+8uv^{3}-4uv^{2}+uv+v^{4}-4v^{3}+8v^{2}-5v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {12}{q^{9/2}}}-{\frac {13}{q^{7/2}}}+{\frac {12}{q^{5/2}}}+q^{3/2}-{\frac {11}{q^{3/2}}}-{\frac {2}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-a^{7}z^{3}-4a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+7a^{5}z^{3}+8a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$3a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+a^{8}z^{6}+4a^{8}z^{4}-4a^{8}z^{2}+a^{8}+5a^{7}z^{7}-13a^{7}z^{5}+25a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+6a^{6}z^{8}-16a^{6}z^{6}+23a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+2a^{5}z^{9}+8a^{5}z^{7}-37a^{5}z^{5}+49a^{5}z^{3}-25a^{5}z+4a^{5}z^{-1}+11a^{4}z^{8}-29a^{4}z^{6}+23a^{4}z^{4}-9a^{4}z^{2}+2a^{4}+2a^{3}z^{9}+7a^{3}z^{7}-35a^{3}z^{5}+35a^{3}z^{3}-15a^{3}z+2a^{3}z^{-1}+5a^{2}z^{8}-11a^{2}z^{6}+2a^{2}z^{4}+a^{2}+4az^{7}-11az^{5}+8az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

3

0

4

1

-3

-2

7

3

4

-4

6

5

-1

-6

7

6

1

-8

5

6

1

-10

4

7

-3

-12

2

6

4

-14

3

-3

-16

2

Integral Khovanov Homology

(db, data source)

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