L11n217

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_8,9,1,10 X_3,12,4,13 X_15,22,16,9 X_17,3,18,2 X_21,4,22,5 X_5,15,6,14 X_13,21,14,20 X_11,16,12,17 X_19,7,20,6 X_7,19,8,18
Gauss code	{1, 5, -3, 6, -7, 10, -11, -2}, {2, -1, -9, 3, -8, 7, -4, 9, -5, 11, -10, 8, -6, 4}

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

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

2

-2

4

2

4

2

-2

0

8

4

-2

6

0

-4

6

0

-6

4

6

2

-8

2

6

-4

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$