L11a292

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_18,10,19,9 X_22,18,9,17 X_8,21,1,22 X_20,13,21,14 X_14,8,15,7 X_16,6,17,5 X_6,16,7,15 X_4,20,5,19
Gauss code	{1, -2, 3, -11, 9, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 10, -9, 5, -4, 11, -7, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-t(1)t(2)^{3}+t(2)^{3}-t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {3}{q^{9/2}}}-6q^{7/2}+{\frac {7}{q^{7/2}}}+10q^{5/2}-{\frac {12}{q^{5/2}}}-14q^{3/2}+{\frac {14}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+16{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-3z^{3}a^{-3}-3a^{3}z-2za^{-3}+az^{7}+z^{7}a^{-1}+4az^{5}+4z^{5}a^{-1}+6az^{3}+5z^{3}a^{-1}+4az+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-9z^{9}a^{-1}-4z^{9}a^{-3}-8a^{2}z^{8}+z^{8}a^{-2}-3z^{8}a^{-4}-4z^{8}-9a^{3}z^{7}+2az^{7}+27z^{7}a^{-1}+15z^{7}a^{-3}-z^{7}a^{-5}-6a^{4}z^{6}+11a^{2}z^{6}+10z^{6}a^{-2}+12z^{6}a^{-4}+15z^{6}-3a^{5}z^{5}+16a^{3}z^{5}+14az^{5}-28z^{5}a^{-1}-19z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-2a^{2}z^{4}-10z^{4}a^{-2}-14z^{4}a^{-4}-5z^{4}+2a^{5}z^{3}-16a^{3}z^{3}-17az^{3}+17z^{3}a^{-1}+12z^{3}a^{-3}-4z^{3}a^{-5}+a^{6}z^{2}-3a^{4}z^{2}-4a^{2}z^{2}+3z^{2}a^{-2}+4z^{2}a^{-4}-z^{2}+6a^{3}z+8az-2za^{-1}-4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

2

-2

8

4

1

3

6

2

-4

4

8

4

2

8

6

-2

0

9

8

1

-2

7

10

3

-4

5

7

-2

-6

2

7

5

-8

1

5

-4

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$