L11a279

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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_20,13,21,14 X_14,6,15,5 X_4,20,5,19 X_18,8,19,7 X_6,16,7,15 X_22,17,9,18 X_16,21,17,22
Gauss code	{1, -2, 3, -7, 6, -9, 8, -4}, {4, -1, 2, -3, 5, -6, 9, -11, 10, -8, 7, -5, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)^{2}t(2)^{4}-t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}+2t(1)t(2)-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {12}{q^{9/2}}}-{\frac {14}{q^{7/2}}}-q^{5/2}+{\frac {13}{q^{5/2}}}+3q^{3/2}-{\frac {12}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-5a^{5}z^{5}-8a^{5}z^{3}-5a^{5}z-a^{5}z^{-1}+a^{3}z^{9}+7a^{3}z^{7}+18a^{3}z^{5}+21a^{3}z^{3}+11a^{3}z+3a^{3}z^{-1}-az^{7}-5az^{5}-8az^{3}-6az-2az^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-5z^{6}a^{8}+6z^{4}a^{8}-2z^{2}a^{8}-6z^{7}a^{7}+9z^{5}a^{7}-6z^{3}a^{7}+za^{7}-6z^{8}a^{6}+12z^{6}a^{6}-12z^{4}a^{6}+6z^{2}a^{6}-a^{6}-5z^{9}a^{5}+13z^{7}a^{5}-17z^{5}a^{5}+14z^{3}a^{5}-6za^{5}+a^{5}z^{-1}-2z^{10}a^{4}-z^{8}a^{4}+18z^{6}a^{4}-27z^{4}a^{4}+16z^{2}a^{4}-3a^{4}-9z^{9}a^{3}+36z^{7}a^{3}-52z^{5}a^{3}+39z^{3}a^{3}-15za^{3}+3a^{3}z^{-1}-2z^{10}a^{2}+2z^{8}a^{2}+13z^{6}a^{2}-20z^{4}a^{2}+10z^{2}a^{2}-3a^{2}-4z^{9}a+16z^{7}a-19z^{5}a+12z^{3}a-7za+2az^{-1}-3z^{8}+12z^{6}-12z^{4}+3z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-4z^{3}a^{-1}+za^{-1}$ (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

2

-2

2

4

1

3

0

4

2

-2

8

4

-4

7

6

-1

-6

7

6

1

-8

5

7

2

-10

4

7

-3

-12

2

5

3

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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