L11a118

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {3uv^{4}-12uv^{3}+14uv^{2}-7uv+u+v^{5}-7v^{4}+14v^{3}-12v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {16}{q^{5/2}}}-{\frac {22}{q^{7/2}}}+{\frac {24}{q^{9/2}}}-{\frac {24}{q^{11/2}}}+{\frac {20}{q^{13/2}}}-{\frac {15}{q^{15/2}}}+{\frac {8}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-6a^{7}z^{3}-8a^{7}z-3a^{7}z^{-1}+3a^{5}z^{5}+6a^{5}z^{3}+6a^{5}z+2a^{5}z^{-1}+a^{3}z^{5}-2a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}-az^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+7z^{5}a^{11}-6z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-5z^{8}a^{10}+8z^{6}a^{10}-z^{4}a^{10}-4z^{2}a^{10}+2a^{10}-5z^{9}a^{9}+2z^{7}a^{9}+13z^{5}a^{9}-17z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-2z^{10}a^{8}-13z^{8}a^{8}+36z^{6}a^{8}-28z^{4}a^{8}+9z^{2}a^{8}-13z^{9}a^{7}+15z^{7}a^{7}+13z^{5}a^{7}-26z^{3}a^{7}+15za^{7}-3a^{7}z^{-1}-2z^{10}a^{6}-20z^{8}a^{6}+52z^{6}a^{6}-44z^{4}a^{6}+15z^{2}a^{6}-2a^{6}-8z^{9}a^{5}+z^{7}a^{5}+22z^{5}a^{5}-26z^{3}a^{5}+11za^{5}-2a^{5}z^{-1}-12z^{8}a^{4}+21z^{6}a^{4}-16z^{4}a^{4}+5z^{2}a^{4}-9z^{7}a^{3}+14z^{5}a^{3}-10z^{3}a^{3}+5za^{3}-a^{3}z^{-1}-4z^{6}a^{2}+4z^{4}a^{2}-z^{5}a+z^{3}a$ (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

3

-3

-2

7

1

6

-4

10

4

-6

12

6

-8

12

10

-2

-10

12

0

-12

9

13

4

-14

6

11

-5

-16

2

9

7

-18

1

6

-5

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-4$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$