L11a312

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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_8,9,1,10 X_22,13,9,14 X_14,8,15,7 X_18,6,19,5 X_20,17,21,18 X_16,21,17,22 X_6,16,7,15 X_4,20,5,19
Gauss code	{1, -2, 3, -11, 7, -10, 6, -4}, {4, -1, 2, -3, 5, -6, 10, -9, 8, -7, 11, -8, 9, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}-2t(2)^{2}t(1)-2t(1)-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+2q^{3/2}-5{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {11}{q^{9/2}}}-{\frac {9}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+2z^{5}a^{5}+6z^{3}a^{5}+3za^{5}-z^{7}a^{3}-4z^{5}a^{3}-5z^{3}a^{3}-3za^{3}+2z^{5}a+7z^{3}a+5za+az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-3a^{9}z^{3}+5a^{8}z^{6}-6a^{8}z^{4}+2a^{8}z^{2}+6a^{7}z^{7}-10a^{7}z^{5}+9a^{7}z^{3}-3a^{7}z+5a^{6}z^{8}-8a^{6}z^{6}+7a^{6}z^{4}-3a^{6}z^{2}+3a^{5}z^{9}-2a^{5}z^{7}-5a^{5}z^{5}+6a^{5}z^{3}-2a^{5}z+a^{4}z^{10}+4a^{4}z^{8}-16a^{4}z^{6}+15a^{4}z^{4}-5a^{4}z^{2}+5a^{3}z^{9}-14a^{3}z^{7}+10a^{3}z^{5}-3a^{3}z^{3}+a^{2}z^{10}+a^{2}z^{8}-11a^{2}z^{6}+9a^{2}z^{4}+2az^{9}-5az^{7}+z^{7}a^{-1}-3az^{5}-5z^{5}a^{-1}+11az^{3}+8z^{3}a^{-1}-6az+az^{-1}-5za^{-1}+a^{-1}z^{-1}+2z^{8}-8z^{6}+8z^{4}-z^{2}-1$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

1

-1

2

4

1

3

0

3

1

-2

7

4

3

-4

7

5

-2

-6

6

5

1

-8

5

7

2

-10

4

6

-2

-12

2

5

3

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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