L11a171

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_18,12,19,11 X_10,4,11,3 X_12,7,13,8 X_16,13,17,14 X_22,17,7,18 X_6,15,1,16 X_4,21,5,22 X_2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {6, -1, 2, -5, 4, -6, 7, -3, 9, -7, 8, -4, 11, -2, 10, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-7t(1)^{2}t(2)^{3}+12t(1)t(2)^{3}-5t(2)^{3}+9t(1)^{2}t(2)^{2}-17t(1)t(2)^{2}+9t(2)^{2}-5t(1)^{2}t(2)+12t(1)t(2)-7t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-5{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {20}{q^{3/2}}}+{\frac {26}{q^{5/2}}}-{\frac {31}{q^{7/2}}}+{\frac {31}{q^{9/2}}}-{\frac {27}{q^{11/2}}}+{\frac {20}{q^{13/2}}}-{\frac {12}{q^{15/2}}}+{\frac {5}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+a^{7}z^{3}-a^{5}z^{7}-2a^{5}z^{5}-a^{5}z^{3}-a^{3}z^{7}-2a^{3}z^{5}-a^{3}z^{3}+a^{3}z^{-1}+az^{5}+az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}+5a^{10}z^{6}-3a^{10}z^{4}+12a^{9}z^{7}-14a^{9}z^{5}+5a^{9}z^{3}+17a^{8}z^{8}-25a^{8}z^{6}+13a^{8}z^{4}-2a^{8}z^{2}+13a^{7}z^{9}-7a^{7}z^{7}-14a^{7}z^{5}+8a^{7}z^{3}+4a^{6}z^{10}+26a^{6}z^{8}-64a^{6}z^{6}+38a^{6}z^{4}-6a^{6}z^{2}+23a^{5}z^{9}-34a^{5}z^{7}+a^{5}z^{5}+6a^{5}z^{3}+4a^{4}z^{10}+19a^{4}z^{8}-54a^{4}z^{6}+34a^{4}z^{4}-6a^{4}z^{2}+10a^{3}z^{9}-10a^{3}z^{7}-9a^{3}z^{5}+8a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+10a^{2}z^{8}-19a^{2}z^{6}+11a^{2}z^{4}-2a^{2}z^{2}+a^{2}+5az^{7}-9az^{5}+5az^{3}+az-az^{-1}+z^{6}-z^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

4

0

7

1

-6

-2

13

4

9

-4

14

8

-6

17

12

5

-8

15

0

-10

12

16

-4

-12

8

15

7

-14

4

12

-8

-16

1

8

7

-18

4

-4

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-4$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{17}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$