L10a169

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Japanese family symbol

Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_14,9,11,10 X_20,13,15,14 X_10,19,5,20 X_18,12,19,11 X_4,17,1,18
Gauss code	{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 6, -8}, {4, -5, 10, -9, 8, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)^{2}(x-1)^{2}}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-11q^{3/2}+15{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {15}{q^{7/2}}}-{\frac {11}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+3az^{3}-z^{3}a^{-1}-a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$a^{7}z^{5}+5a^{6}z^{6}-4a^{6}z^{4}+11a^{5}z^{7}-17a^{5}z^{5}+8a^{5}z^{3}-a^{5}z^{-3}+a^{5}z^{-1}+11a^{4}z^{8}-12a^{4}z^{6}+3a^{4}z^{-2}-2a^{4}+4a^{3}z^{9}+19a^{3}z^{7}-48a^{3}z^{5}+z^{5}a^{-3}+24a^{3}z^{3}-3a^{3}z^{-3}+a^{3}z+22a^{2}z^{8}-34a^{2}z^{6}+5z^{6}a^{-2}+8a^{2}z^{4}-4z^{4}a^{-2}+6a^{2}z^{-2}-3a^{2}+4az^{9}+19az^{7}+11z^{7}a^{-1}-48az^{5}-17z^{5}a^{-1}+24az^{3}+8z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}+az+a^{-1}z^{-1}+11z^{8}-12z^{6}+3z^{-2}-2$ (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

χ

8

1

6

4

-4

4

7

1

6

2

8

4

-4

0

14

7

-2

10

12

2

-4

12

10

2

-6

7

14

7

-8

4

8

-4

-10

1

7

6

-12

4

-4

-14

1

Integral Khovanov Homology

(db, data source)

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