L11a337

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_22,5,9,6 X_14,3,15,4 X_4,15,5,16 X_16,7,17,8 X_20,17,21,18 X_18,11,19,12 X_12,19,13,20 X_2,9,3,10 X_8,13,1,14 X_6,21,7,22
Gauss code	{1, -9, 3, -4, 2, -11, 5, -10}, {9, -1, 7, -8, 10, -3, 4, -5, 6, -7, 8, -6, 11, -2}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(2)^{5}-4t(1)^{2}t(2)^{4}+5t(1)t(2)^{4}-t(2)^{4}-2t(1)^{3}t(2)^{3}+9t(1)^{2}t(2)^{3}-10t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-10t(1)^{2}t(2)^{2}+9t(1)t(2)^{2}-2t(2)^{2}-t(1)^{3}t(2)+5t(1)^{2}t(2)-4t(1)t(2)-t(1)^{2}}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {9}{q^{23/2}}}-{\frac {15}{q^{21/2}}}+{\frac {19}{q^{19/2}}}-{\frac {22}{q^{17/2}}}+{\frac {22}{q^{15/2}}}-{\frac {19}{q^{13/2}}}+{\frac {13}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+4z^{3}a^{11}+7za^{11}+3a^{11}z^{-1}-3z^{5}a^{9}-7z^{3}a^{9}-5za^{9}-2a^{9}z^{-1}-3z^{5}a^{7}-7z^{3}a^{7}-4za^{7}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+2z^{4}a^{16}-z^{2}a^{16}-4z^{7}a^{15}+9z^{5}a^{15}-6z^{3}a^{15}+za^{15}-7z^{8}a^{14}+15z^{6}a^{14}-8z^{4}a^{14}+2z^{2}a^{14}-a^{14}-6z^{9}a^{13}+6z^{7}a^{13}+8z^{5}a^{13}-6z^{3}a^{13}+a^{13}z^{-1}-2z^{10}a^{12}-12z^{8}a^{12}+36z^{6}a^{12}-31z^{4}a^{12}+15z^{2}a^{12}-3a^{12}-12z^{9}a^{11}+21z^{7}a^{11}-16z^{5}a^{11}+19z^{3}a^{11}-14za^{11}+3a^{11}z^{-1}-2z^{10}a^{10}-12z^{8}a^{10}+31z^{6}a^{10}-28z^{4}a^{10}+12z^{2}a^{10}-3a^{10}-6z^{9}a^{9}+5z^{7}a^{9}-4z^{5}a^{9}+8z^{3}a^{9}-8za^{9}+2a^{9}z^{-1}-7z^{8}a^{8}+8z^{6}a^{8}-3z^{4}a^{8}-z^{2}a^{8}-6z^{7}a^{7}+10z^{5}a^{7}-9z^{3}a^{7}+4za^{7}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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

3

1

-2

-8

5

-10

8

3

-5

-12

11

5

6

-14

11

8

-3

-16

11

0

-18

8

11

3

-20

7

11

-4

-22

3

9

6

-24

1

6

-5

-26

3

-28

1

-1

Integral Khovanov Homology

(db, data source)

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