L11a296

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-t(1)t(2)^{3}+t(2)^{3}+3t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}-q^{7/2}+{\frac {11}{q^{7/2}}}+3q^{5/2}-{\frac {14}{q^{5/2}}}-6q^{3/2}+{\frac {14}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+9{\sqrt {q}}-{\frac {13}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+5a^{3}z^{5}+4az^{5}-z^{5}a^{-1}-4a^{5}z^{3}+10a^{3}z^{3}+4az^{3}-3z^{3}a^{-1}-5a^{5}z+9a^{3}z-2az-2za^{-1}-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-6a^{3}z^{9}-3az^{9}-3a^{6}z^{8}-3a^{4}z^{8}-5a^{2}z^{8}-5z^{8}-2a^{7}z^{7}+10a^{5}z^{7}+19a^{3}z^{7}+2az^{7}-5z^{7}a^{-1}-a^{8}z^{6}+9a^{6}z^{6}+14a^{4}z^{6}+18a^{2}z^{6}-3z^{6}a^{-2}+11z^{6}+6a^{7}z^{5}-21a^{5}z^{5}-35a^{3}z^{5}+5az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+4a^{8}z^{4}-10a^{6}z^{4}-23a^{4}z^{4}-25a^{2}z^{4}+6z^{4}a^{-2}-10z^{4}-4a^{7}z^{3}+27a^{5}z^{3}+35a^{3}z^{3}-8az^{3}-10z^{3}a^{-1}+2z^{3}a^{-3}-4a^{8}z^{2}+7a^{6}z^{2}+18a^{4}z^{2}+11a^{2}z^{2}-z^{2}a^{-2}+3z^{2}-10a^{5}z-17a^{3}z-3az+4za^{-1}-a^{6}-3a^{4}-3a^{2}+a^{5}z^{-1}+3a^{3}z^{-1}+2az^{-1}$ (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

8

1

6

2

-2

4

1

3

2

5

2

-3

0

8

4

-2

8

7

-1

-4

6

0

-6

5

8

3

-8

4

6

-2

-10

1

5

4

-12

1

4

-3

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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