L11n110

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-4uv^{4}+7uv^{3}-6uv^{2}+uv+v^{4}-6v^{3}+7v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+12{\sqrt {q}}-{\frac {13}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+5az^{5}-2z^{5}a^{-1}-4a^{3}z^{3}+11az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-8a^{3}z+12az-9za^{-1}+2za^{-3}+2a^{5}z^{-1}-5a^{3}z^{-1}+6az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$6a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+3a^{4}z^{6}+z^{6}a^{-4}-3z^{4}a^{-4}+a^{4}z^{2}+3z^{2}a^{-4}-a^{4}-a^{-4}+7a^{3}z^{7}+3z^{7}a^{-3}-17a^{3}z^{5}-9z^{5}a^{-3}+29a^{3}z^{3}+9z^{3}a^{-3}-21a^{3}z-4za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+5a^{2}z^{8}+3z^{8}a^{-2}-5a^{2}z^{6}-3z^{6}a^{-2}-9z^{4}a^{-2}+3a^{2}z^{2}+11z^{2}a^{-2}-a^{2}-3a^{-2}+az^{9}+z^{9}a^{-1}+14az^{7}+10z^{7}a^{-1}-44az^{5}-36z^{5}a^{-1}+51az^{3}+37z^{3}a^{-1}-29az-19za^{-1}+6az^{-1}+4a^{-1}z^{-1}+8z^{8}-12z^{6}-6z^{4}+10z^{2}-3$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

4

1

-3

4

6

2

4

2

6

4

-2

0

7

6

1

-2

6

7

1

-4

4

6

-2

-6

2

6

4

-8

1

4

-3

-10

3

Integral Khovanov Homology

(db, data source)

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