L11a145

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_18,12,19,11 X_12,6,13,5 X_4,19,5,20 X_14,7,15,8 X_20,13,21,14 X_22,15,7,16 X_16,21,17,22 X_6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -10, 11, -4, 6, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-5t(1)^{2}t(2)^{3}+7t(1)t(2)^{3}-2t(2)^{3}+5t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+5t(2)^{2}-2t(1)^{2}t(2)+7t(1)t(2)-5t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {17}{q^{9/2}}}-{\frac {18}{q^{7/2}}}+{\frac {15}{q^{5/2}}}+q^{3/2}-{\frac {12}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {11}{q^{13/2}}}-{\frac {15}{q^{11/2}}}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+2za^{7}-z^{7}a^{5}-4z^{5}a^{5}-6z^{3}a^{5}-4za^{5}-z^{7}a^{3}-3z^{5}a^{3}-z^{3}a^{3}+2za^{3}+a^{3}z^{-1}+z^{5}a+2z^{3}a-za-az^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-3z^{6}a^{10}+6z^{4}a^{10}-2z^{2}a^{10}-5z^{7}a^{9}+10z^{5}a^{9}-6z^{3}a^{9}+za^{9}-6z^{8}a^{8}+13z^{6}a^{8}-13z^{4}a^{8}+4z^{2}a^{8}-5z^{9}a^{7}+10z^{7}a^{7}-12z^{5}a^{7}+5z^{3}a^{7}-za^{7}-2z^{10}a^{6}-4z^{8}a^{6}+19z^{6}a^{6}-28z^{4}a^{6}+12z^{2}a^{6}-10z^{9}a^{5}+29z^{7}a^{5}-38z^{5}a^{5}+23z^{3}a^{5}-5za^{5}-2z^{10}a^{4}-4z^{8}a^{4}+20z^{6}a^{4}-21z^{4}a^{4}+8z^{2}a^{4}-5z^{9}a^{3}+10z^{7}a^{3}-4z^{5}a^{3}+4z^{3}a^{3}-4za^{3}+a^{3}z^{-1}-6z^{8}a^{2}+16z^{6}a^{2}-10z^{4}a^{2}+2z^{2}a^{2}-a^{2}-4z^{7}a+11z^{5}a-6z^{3}a-za+az^{-1}-z^{6}+2z^{4}$ (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

4

1

-1

2

3

0

4

1

-3

-2

8

3

5

-4

8

5

-3

-6

10

7

3

-8

8

9

1

-10

7

9

-2

-12

4

8

4

-14

2

7

-5

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$