L11a170

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_12,7,13,8 X_22,15,7,16 X_14,6,15,5 X_6,14,1,13 X_16,21,17,22 X_18,10,19,9 X_20,11,21,12 X_4,18,5,17 X_2,19,3,20
Gauss code	{1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, 4, -7, 10, -8, 11, -9, 7, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-5u^{2}v^{3}+8u^{2}v^{2}-6u^{2}v+2u^{2}-3uv^{4}+11uv^{3}-15uv^{2}+11uv-3u+2v^{4}-6v^{3}+8v^{2}-5v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {10}{q^{9/2}}}-5q^{7/2}+{\frac {18}{q^{7/2}}}+11q^{5/2}-{\frac {24}{q^{5/2}}}-19q^{3/2}+{\frac {28}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+24{\sqrt {q}}-{\frac {29}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+2az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+az^{3}-2z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-a^{3}z-az+2za^{-1}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+9a^{5}z^{7}-12a^{5}z^{5}+7a^{5}z^{3}-a^{5}z+13a^{4}z^{8}-21a^{4}z^{6}+z^{6}a^{-4}+17a^{4}z^{4}-z^{4}a^{-4}-6a^{4}z^{2}+11a^{3}z^{9}-11a^{3}z^{7}+5z^{7}a^{-3}-9z^{5}a^{-3}+2a^{3}z^{3}+4z^{3}a^{-3}+a^{3}z+za^{-3}-a^{-3}z^{-1}+4a^{2}z^{10}+17a^{2}z^{8}+10z^{8}a^{-2}-48a^{2}z^{6}-20z^{6}a^{-2}+38a^{2}z^{4}+10z^{4}a^{-2}-12a^{2}z^{2}-z^{2}a^{-2}+a^{-2}+21az^{9}+10z^{9}a^{-1}-39az^{7}-14z^{7}a^{-1}+21az^{5}-z^{5}a^{-1}-10az^{3}+5az+4za^{-1}-a^{-1}z^{-1}+4z^{10}+14z^{8}-44z^{6}+28z^{4}-6z^{2}$ (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

4

5

χ

10

1

-1

8

4

6

7

1

-6

4

12

4

8

2

13

8

-5

0

16

11

5

-2

13

14

1

-4

11

15

-4

-6

7

13

6

-8

3

11

-8

-10

1

7

6

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{16}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$