L11a29

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(5t(2)^{2}-8t(2)+5\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$22q^{9/2}-21q^{7/2}+14q^{5/2}-8q^{3/2}+q^{21/2}-4q^{19/2}+10q^{17/2}-15q^{15/2}+21q^{13/2}-24q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}+z^{3}a^{-7}+za^{-7}-a^{-7}z^{-1}-3z^{5}a^{-5}-6z^{3}a^{-5}-5za^{-5}-2a^{-5}z^{-1}-z^{5}a^{-3}+z^{3}a^{-3}+3za^{-3}+2a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-6}-2z^{10}a^{-8}-6z^{9}a^{-5}-13z^{9}a^{-7}-7z^{9}a^{-9}-8z^{8}a^{-4}-16z^{8}a^{-6}-16z^{8}a^{-8}-8z^{8}a^{-10}-6z^{7}a^{-3}-2z^{7}a^{-5}+17z^{7}a^{-7}+9z^{7}a^{-9}-4z^{7}a^{-11}-3z^{6}a^{-2}+11z^{6}a^{-4}+42z^{6}a^{-6}+48z^{6}a^{-8}+19z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+9z^{5}a^{-3}+23z^{5}a^{-5}+8z^{5}a^{-7}+3z^{5}a^{-9}+8z^{5}a^{-11}+4z^{4}a^{-2}-7z^{4}a^{-4}-37z^{4}a^{-6}-44z^{4}a^{-8}-16z^{4}a^{-10}+2z^{4}a^{-12}+2z^{3}a^{-1}-9z^{3}a^{-3}-31z^{3}a^{-5}-19z^{3}a^{-7}-3z^{3}a^{-9}-4z^{3}a^{-11}-z^{2}a^{-2}-2z^{2}a^{-4}+14z^{2}a^{-6}+25z^{2}a^{-8}+9z^{2}a^{-10}-z^{2}a^{-12}-za^{-1}+7za^{-3}+15za^{-5}+9za^{-7}+2za^{-9}+2a^{-4}-4a^{-6}-9a^{-8}-4a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-7}z^{-1}+a^{-9}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

3

18

7

1

-6

16

8

3

5

14

13

7

-6

12

11

8

3

10

11

13

2

8

10

11

-1

6

4

11

7

4

10

-6

2

1

6

5

0

2

-2

1

Integral Khovanov Homology

(db, data source)

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