L11a538

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)(t(4)-1)\left(t(1)t(4)^{2}+t(2)t(4)^{2}-t(4)^{2}-3t(1)t(4)+2t(1)t(2)t(4)-3t(2)t(4)+2t(4)+t(1)-t(1)t(2)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	${\frac {19}{q^{9/2}}}-{\frac {20}{q^{7/2}}}+{\frac {13}{q^{5/2}}}-{\frac {8}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {14}{q^{15/2}}}+{\frac {17}{q^{13/2}}}-{\frac {22}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+4a^{9}z^{-1}+a^{9}z^{-3}-6z^{3}a^{7}-11za^{7}-9a^{7}z^{-1}-3a^{7}z^{-3}+3z^{5}a^{5}+8z^{3}a^{5}+12za^{5}+10a^{5}z^{-1}+3a^{5}z^{-3}+z^{5}a^{3}-z^{3}a^{3}-4za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{12}z^{6}-3a^{12}z^{4}+3a^{12}z^{2}-a^{12}+3a^{11}z^{7}-8a^{11}z^{5}+9a^{11}z^{3}-6a^{11}z+2a^{11}z^{-1}+4a^{10}z^{8}-4a^{10}z^{6}-8a^{10}z^{4}+14a^{10}z^{2}-6a^{10}+3a^{9}z^{9}+7a^{9}z^{7}-34a^{9}z^{5}+43a^{9}z^{3}-a^{9}z^{-3}-30a^{9}z+11a^{9}z^{-1}+a^{8}z^{10}+12a^{8}z^{8}-21a^{8}z^{6}-8a^{8}z^{4}+27a^{8}z^{2}+3a^{8}z^{-2}-18a^{8}+7a^{7}z^{9}+9a^{7}z^{7}-59a^{7}z^{5}+79a^{7}z^{3}-3a^{7}z^{-3}-52a^{7}z+18a^{7}z^{-1}+a^{6}z^{10}+15a^{6}z^{8}-26a^{6}z^{6}-a^{6}z^{4}+27a^{6}z^{2}+6a^{6}z^{-2}-21a^{6}+4a^{5}z^{9}+11a^{5}z^{7}-44a^{5}z^{5}+59a^{5}z^{3}-3a^{5}z^{-3}-40a^{5}z+14a^{5}z^{-1}+7a^{4}z^{8}-7a^{4}z^{6}-2a^{4}z^{4}+12a^{4}z^{2}+3a^{4}z^{-2}-9a^{4}+6a^{3}z^{7}-10a^{3}z^{5}+12a^{3}z^{3}-a^{3}z^{-3}-11a^{3}z+5a^{3}z^{-1}+3a^{2}z^{6}-4a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

4

-5

-6

11

4

7

-8

8

9

1

-10

14

11

3

-12

9

14

5

-14

5

8

-3

-16

2

9

7

-18

1

5

-4

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-4$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{14}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$