L11a97

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_20,17,21,18 X_18,13,19,14 X_14,19,15,20 X_4,21,1,22 X_10,5,11,6 X_12,3,13,4 X_22,11,5,12 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, 4, -5, 11, -2, 3, -4, 5, -3, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+6t(1)t(2)^{4}-6t(2)^{4}-13t(1)t(2)^{3}+13t(2)^{3}+13t(1)t(2)^{2}-13t(2)^{2}-6t(1)t(2)+6t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {16}{q^{11/2}}}-{\frac {23}{q^{13/2}}}+{\frac {25}{q^{15/2}}}-{\frac {25}{q^{17/2}}}+{\frac {22}{q^{19/2}}}-{\frac {15}{q^{21/2}}}+{\frac {9}{q^{23/2}}}-{\frac {4}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}+3z^{3}a^{11}+2za^{11}-2a^{11}z^{-1}-2z^{5}a^{9}+7za^{9}+5a^{9}z^{-1}-4z^{5}a^{7}-11z^{3}a^{7}-10za^{7}-3a^{7}z^{-1}-z^{5}a^{5}-z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+2z^{4}a^{16}-z^{2}a^{16}-4z^{7}a^{15}+9z^{5}a^{15}-6z^{3}a^{15}+za^{15}-7z^{8}a^{14}+15z^{6}a^{14}-11z^{4}a^{14}+4z^{2}a^{14}-a^{14}-6z^{9}a^{13}+3z^{7}a^{13}+14z^{5}a^{13}-13z^{3}a^{13}+2za^{13}-2z^{10}a^{12}-17z^{8}a^{12}+48z^{6}a^{12}-41z^{4}a^{12}+14z^{2}a^{12}-14z^{9}a^{11}+15z^{7}a^{11}+12z^{5}a^{11}-14z^{3}a^{11}+4za^{11}-2a^{11}z^{-1}-2z^{10}a^{10}-22z^{8}a^{10}+52z^{6}a^{10}-34z^{4}a^{10}+2z^{2}a^{10}+5a^{10}-8z^{9}a^{9}-2z^{7}a^{9}+25z^{5}a^{9}-24z^{3}a^{9}+13za^{9}-5a^{9}z^{-1}-12z^{8}a^{8}+16z^{6}a^{8}-3z^{4}a^{8}-7z^{2}a^{8}+5a^{8}-10z^{7}a^{7}+17z^{5}a^{7}-16z^{3}a^{7}+10za^{7}-3a^{7}z^{-1}-4z^{6}a^{6}+3z^{4}a^{6}-z^{5}a^{5}+z^{3}a^{5}$ (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

4

1

-3

-8

7

-10

9

4

-5

-12

14

7

-14

12

10

-2

-16

13

0

-18

9

12

3

-20

6

13

-7

-22

3

9

6

-24

1

6

-5

-26

3

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-8$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-7$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-6$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-5$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$