L11a24

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_18,11,19,12 X_22,19,5,20 X_20,15,21,16 X_16,21,17,22 X_12,17,13,18 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3uv^{4}-10uv^{3}+12uv^{2}-7uv+u+v^{5}-7v^{4}+12v^{3}-10v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {21}{q^{9/2}}}-{\frac {22}{q^{11/2}}}+{\frac {18}{q^{13/2}}}-{\frac {14}{q^{15/2}}}+{\frac {8}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-6a^{7}z^{3}-9a^{7}z-3a^{7}z^{-1}+3a^{5}z^{5}+7a^{5}z^{3}+7a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+7z^{5}a^{11}-6z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-5z^{8}a^{10}+9z^{6}a^{10}-3z^{4}a^{10}-2z^{2}a^{10}+2a^{10}-4z^{9}a^{9}-2z^{7}a^{9}+21z^{5}a^{9}-21z^{3}a^{9}+11za^{9}-3a^{9}z^{-1}-z^{10}a^{8}-16z^{8}a^{8}+42z^{6}a^{8}-34z^{4}a^{8}+12z^{2}a^{8}-9z^{9}a^{7}+2z^{7}a^{7}+31z^{5}a^{7}-36z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-z^{10}a^{6}-19z^{8}a^{6}+48z^{6}a^{6}-45z^{4}a^{6}+18z^{2}a^{6}-2a^{6}-5z^{9}a^{5}-5z^{7}a^{5}+27z^{5}a^{5}-31z^{3}a^{5}+13za^{5}-2a^{5}z^{-1}-8z^{8}a^{4}+13z^{6}a^{4}-13z^{4}a^{4}+6z^{2}a^{4}-6z^{7}a^{3}+9z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-3z^{6}a^{2}+4z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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

2

1

0

2

-2

6

1

5

-4

9

3

-6

10

5

-8

11

9

-2

-10

11

10

1

-12

8

12

4

-14

6

10

-4

-16

2

8

6

-18

1

6

-5

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$