L11a259

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-3t(1)^{2}t(2)^{4}+2t(1)t(2)^{4}-t(2)^{4}-t(1)^{3}t(2)^{3}+3t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}+t(1)^{3}t(2)^{2}-3t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-t(2)^{2}-t(1)^{3}t(2)+2t(1)^{2}t(2)-3t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}+{\frac {7}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {4}{q^{19/2}}}-{\frac {7}{q^{17/2}}}+{\frac {9}{q^{15/2}}}-{\frac {11}{q^{13/2}}}+{\frac {10}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-4z^{3}a^{9}-4za^{9}-a^{9}z^{-1}+z^{7}a^{7}+5z^{5}a^{7}+9z^{3}a^{7}+8za^{7}+3a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+3z^{3}a^{5}-2za^{5}-2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+2a^{13}z^{5}-3a^{13}z^{3}+3a^{12}z^{6}-5a^{12}z^{4}+3a^{12}z^{2}+4a^{11}z^{7}-10a^{11}z^{5}+13a^{11}z^{3}-4a^{11}z+4a^{10}z^{8}-11a^{10}z^{6}+14a^{10}z^{4}-4a^{10}z^{2}+a^{10}+3a^{9}z^{9}-8a^{9}z^{7}+9a^{9}z^{5}-3a^{9}z^{3}+a^{9}z-a^{9}z^{-1}+a^{8}z^{10}+2a^{8}z^{8}-14a^{8}z^{6}+20a^{8}z^{4}-14a^{8}z^{2}+3a^{8}+5a^{7}z^{9}-19a^{7}z^{7}+29a^{7}z^{5}-30a^{7}z^{3}+15a^{7}z-3a^{7}z^{-1}+a^{6}z^{10}-8a^{6}z^{6}+8a^{6}z^{4}-6a^{6}z^{2}+3a^{6}+2a^{5}z^{9}-6a^{5}z^{7}+3a^{5}z^{5}-3a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-8a^{4}z^{6}+8a^{4}z^{4}-a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-4a^{3}z$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

4

1

3

-6

4

2

-2

-8

5

3

2

-10

5

4

-1

-12

6

5

1

-14

4

6

2

-16

3

5

-2

-18

1

4

3

-20

1

3

-2

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$