L11a291

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}-2t(1)^{2}t(2)^{2}+2t(1)t(2)^{2}-2t(2)^{2}+t(1)^{2}t(2)-3t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{11/2}+5q^{9/2}-10q^{7/2}+15q^{5/2}-21q^{3/2}+22{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-2a^{3}z^{3}-z^{3}a^{-3}+2za^{-3}+az^{7}+z^{7}a^{-1}+3az^{5}+2z^{5}a^{-1}+3az^{3}-2z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-4z^{10}a^{-2}-4z^{10}-11az^{9}-19z^{9}a^{-1}-8z^{9}a^{-3}-14a^{2}z^{8}-2z^{8}a^{-2}-5z^{8}a^{-4}-11z^{8}-12a^{3}z^{7}+16az^{7}+54z^{7}a^{-1}+25z^{7}a^{-3}-z^{7}a^{-5}-8a^{4}z^{6}+23a^{2}z^{6}+35z^{6}a^{-2}+15z^{6}a^{-4}+51z^{6}-4a^{5}z^{5}+14a^{3}z^{5}+3az^{5}-37z^{5}a^{-1}-20z^{5}a^{-3}+2z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-10a^{2}z^{4}-34z^{4}a^{-2}-11z^{4}a^{-4}-40z^{4}+2a^{5}z^{3}-4a^{3}z^{3}-9az^{3}-z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-a^{2}z^{2}+5z^{2}a^{-2}+5z^{2}+4az+8za^{-1}+4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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

4

-4

8

6

1

5

6

9

4

-5

4

12

6

2

10

9

-1

0

13

12

1

-2

9

12

3

-4

5

11

-6

3

9

6

-8

1

5

-4

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$