L11n51

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_18,14,19,13 X_9,17,10,16 X_17,9,18,8 X_22,20,5,19 X_20,15,21,16 X_14,21,15,22 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, 4, -9, 8, 5, -6, -4, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-2v^{3}-2v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{13/2}-4q^{11/2}+6q^{9/2}-7q^{7/2}+8q^{5/2}-8q^{3/2}+6{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-8z^{3}a^{-1}+9z^{3}a^{-3}-4z^{3}a^{-5}+3az-9za^{-1}+9za^{-3}-4za^{-5}+za^{-7}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{2}a^{-8}+4z^{3}a^{-7}-za^{-7}+2z^{6}a^{-6}-z^{4}a^{-6}+z^{2}a^{-6}-a^{-6}+5z^{7}a^{-5}-15z^{5}a^{-5}+16z^{3}a^{-5}-6za^{-5}+a^{-5}z^{-1}+4z^{8}a^{-4}-10z^{6}a^{-4}+3z^{4}a^{-4}+4z^{2}a^{-4}-3a^{-4}+z^{9}a^{-3}+6z^{7}a^{-3}-32z^{5}a^{-3}+37z^{3}a^{-3}-17za^{-3}+3a^{-3}z^{-1}+6z^{8}a^{-2}-20z^{6}a^{-2}+12z^{4}a^{-2}+4z^{2}a^{-2}-3a^{-2}+z^{9}a^{-1}+az^{7}+2z^{7}a^{-1}-5az^{5}-22z^{5}a^{-1}+9az^{3}+34z^{3}a^{-1}-7az-19za^{-1}+2az^{-1}+4a^{-1}z^{-1}+2z^{8}-8z^{6}+8z^{4}-2$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

14

1

-1

12

3

10

3

1

-2

8

4

3

1

6

4

3

-1

4

0

2

4

6

2

0

1

2

-1

-2

1

4

3

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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