L11a276

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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_14,5,15,6 X_18,7,19,8 X_22,20,9,19 X_20,16,21,15 X_16,22,17,21 X_8,9,1,10 X_4,13,5,14 X_6,17,7,18
Gauss code	{1, -2, 3, -10, 4, -11, 5, -9}, {9, -1, 2, -3, 10, -4, 7, -8, 11, -5, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{3}t(2)^{5}-2t(1)^{3}t(2)^{4}+2t(1)^{2}t(2)^{4}+2t(1)^{3}t(2)^{3}-3t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-2t(2)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {2}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {5}{q^{15/2}}}-{\frac {6}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{5}z^{9}+a^{7}z^{7}-8a^{5}z^{7}+a^{3}z^{7}+6a^{7}z^{5}-24a^{5}z^{5}+6a^{3}z^{5}+12a^{7}z^{3}-34a^{5}z^{3}+11a^{3}z^{3}+10a^{7}z-21a^{5}z+7a^{3}z+2a^{7}z^{-1}-3a^{5}z^{-1}+a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+z^{2}a^{12}-3z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-3z^{6}a^{10}+3z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+5z^{5}a^{9}-2z^{3}a^{9}+za^{9}-3z^{8}a^{8}+8z^{6}a^{8}-6z^{4}a^{8}+4z^{2}a^{8}-3z^{9}a^{7}+13z^{7}a^{7}-22z^{5}a^{7}+24z^{3}a^{7}-12za^{7}+2a^{7}z^{-1}-z^{10}a^{6}+16z^{6}a^{6}-30z^{4}a^{6}+19z^{2}a^{6}-3a^{6}-5z^{9}a^{5}+28z^{7}a^{5}-53z^{5}a^{5}+48z^{3}a^{5}-23za^{5}+3a^{5}z^{-1}-z^{10}a^{4}+2z^{8}a^{4}+11z^{6}a^{4}-30z^{4}a^{4}+20z^{2}a^{4}-3a^{4}-2z^{9}a^{3}+12z^{7}a^{3}-23z^{5}a^{3}+18z^{3}a^{3}-7za^{3}+a^{3}z^{-1}-z^{8}a^{2}+6z^{6}a^{2}-11z^{4}a^{2}+7z^{2}a^{2}-a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

2

1

-1

0

1

-2

2

1

-1

-4

3

1

2

-6

3

0

-8

4

2

-10

3

0

-12

3

4

-1

-14

2

3

1

-16

2

3

-1

-18

2

-20

1

2

-1

-22

1

Integral Khovanov Homology

(db, data source)

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