L11a517

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_12,14,7,13 X₂₇₃₈ X_22,10,13,9 X_6,22,1,21 X_20,16,21,15 X_16,5,17,6 X_18,11,19,12 X_10,17,11,18 X_4,19,5,20
Gauss code	{1, -4, 2, -11, 8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, 7, -8, 10, -9, 11, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+3t(1)t(3)t(2)-3t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)-1\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-5q^{-5}+4q^{4}+12q^{-4}-10q^{3}-18q^{-3}+18q^{2}+26q^{-2}-24q-28q^{-1}+29$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-2a^{2}z^{6}-z^{6}a^{-2}+5z^{6}+a^{4}z^{4}-6a^{2}z^{4}-3z^{4}a^{-2}+11z^{4}+a^{4}z^{2}-7a^{2}z^{2}-4z^{2}a^{-2}+10z^{2}+a^{4}-4a^{2}-a^{-2}+4+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$4a^{2}z^{10}+4z^{10}+11a^{3}z^{9}+22az^{9}+11z^{9}a^{-1}+11a^{4}z^{8}+17a^{2}z^{8}+13z^{8}a^{-2}+19z^{8}+5a^{5}z^{7}-18a^{3}z^{7}-42az^{7}-10z^{7}a^{-1}+9z^{7}a^{-3}+a^{6}z^{6}-24a^{4}z^{6}-58a^{2}z^{6}-21z^{6}a^{-2}+4z^{6}a^{-4}-58z^{6}-8a^{5}z^{5}+3a^{3}z^{5}+20az^{5}-4z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-a^{6}z^{4}+15a^{4}z^{4}+50a^{2}z^{4}+18z^{4}a^{-2}-4z^{4}a^{-4}+56z^{4}+2a^{5}z^{3}+2a^{3}z^{3}+8z^{3}a^{-1}+7z^{3}a^{-3}-z^{3}a^{-5}-4a^{4}z^{2}-21a^{2}z^{2}-9z^{2}a^{-2}+z^{2}a^{-4}-27z^{2}-2a^{3}z-4az-3za^{-1}-za^{-3}+3a^{4}+7a^{2}+2a^{-2}+7+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

1

-6

5

11

3

8

3

13

7

-6

1

16

11

5

-1

14

15

1

-3

12

14

-2

-5

8

16

8

-7

4

10

-6

-9

1

8

7

-11

4

-4

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{16}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$