L11a36

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3uv^{4}-6uv^{3}+7uv^{2}-5uv+2u+2v^{5}-5v^{4}+7v^{3}-6v^{2}+3v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {3}{q^{25/2}}}+{\frac {7}{q^{23/2}}}-{\frac {10}{q^{21/2}}}+{\frac {13}{q^{19/2}}}-{\frac {15}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {13}{q^{13/2}}}+{\frac {8}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+3z^{3}a^{11}+5za^{11}+a^{11}z^{-1}-2z^{5}a^{9}-4z^{3}a^{9}+2a^{9}z^{-1}-2z^{5}a^{7}-5z^{3}a^{7}-4za^{7}-2a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-2za^{5}$ (db)
Kauffman polynomial	$a^{16}z^{6}-3a^{16}z^{4}+3a^{16}z^{2}-a^{16}+3a^{15}z^{7}-8a^{15}z^{5}+5a^{15}z^{3}-a^{15}z+4a^{14}z^{8}-8a^{14}z^{6}+a^{14}z^{2}+3a^{13}z^{9}-2a^{13}z^{7}-9a^{13}z^{5}+6a^{13}z^{3}-a^{13}z^{-1}+a^{12}z^{10}+6a^{12}z^{8}-21a^{12}z^{6}+22a^{12}z^{4}-12a^{12}z^{2}+3a^{12}+6a^{11}z^{9}-13a^{11}z^{7}+10a^{11}z^{5}-a^{11}z^{3}+a^{11}z-a^{11}z^{-1}+a^{10}z^{10}+5a^{10}z^{8}-17a^{10}z^{6}+21a^{10}z^{4}-5a^{10}z^{2}+3a^{9}z^{9}-5a^{9}z^{7}+4a^{9}z^{5}+7a^{9}z^{3}-8a^{9}z+2a^{9}z^{-1}+3a^{8}z^{8}-3a^{8}z^{6}-2a^{8}z^{4}+6a^{8}z^{2}-3a^{8}+3a^{7}z^{7}-6a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+2a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-3a^{5}z^{3}+2a^{5}z$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

2

1

-1

-8

3

-10

5

2

-3

-12

8

3

5

-14

7

6

-1

-16

8

7

1

-18

5

7

2

-20

5

8

-3

-22

2

5

3

-24

1

5

-4

-26

2

-28

1

-1

Integral Khovanov Homology

(db, data source)

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