L11a150

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-7u^{2}v^{3}+9u^{2}v^{2}-5u^{2}v+u^{2}-3uv^{4}+12uv^{3}-19uv^{2}+12uv-3u+v^{4}-5v^{3}+9v^{2}-7v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {20}{q^{9/2}}}-q^{7/2}+{\frac {27}{q^{7/2}}}+5q^{5/2}-{\frac {32}{q^{5/2}}}-12q^{3/2}+{\frac {31}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {12}{q^{11/2}}}+20{\sqrt {q}}-{\frac {28}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-a^{5}z^{3}+a^{3}z^{7}+2a^{3}z^{5}+a^{3}z^{3}+a^{3}z^{-1}+az^{7}+2az^{5}-z^{5}a^{-1}+az^{3}-z^{3}a^{-1}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-a^{8}z^{4}+5a^{7}z^{7}-8a^{7}z^{5}+3a^{7}z^{3}+11a^{6}z^{8}-22a^{6}z^{6}+13a^{6}z^{4}-2a^{6}z^{2}+12a^{5}z^{9}-19a^{5}z^{7}+4a^{5}z^{5}+2a^{5}z^{3}+5a^{4}z^{10}+16a^{4}z^{8}-53a^{4}z^{6}+36a^{4}z^{4}-6a^{4}z^{2}+26a^{3}z^{9}-47a^{3}z^{7}+18a^{3}z^{5}+z^{5}a^{-3}+a^{3}z-a^{3}z^{-1}+5a^{2}z^{10}+22a^{2}z^{8}-59a^{2}z^{6}+5z^{6}a^{-2}+36a^{2}z^{4}-3z^{4}a^{-2}-6a^{2}z^{2}+a^{2}+14az^{9}-11az^{7}+12z^{7}a^{-1}-9az^{5}-14z^{5}a^{-1}+6az^{3}+5z^{3}a^{-1}+az-az^{-1}+17z^{8}-24z^{6}+11z^{4}-2z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

4

-4

4

8

1

7

2

12

4

-8

0

16

8

-2

16

13

-3

-4

16

15

1

-6

12

17

5

-8

8

15

-7

-10

4

12

8

-12

1

8

-7

-14

4

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-2$	${\mathbb {Z} }^{17}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{16}$
$r=-1$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{16}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{16}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$