L11n314

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,12,4,13 X_7,17,8,16 X_9,20,10,21 X_11,18,12,19 X_19,22,20,11 X_15,9,16,8 X_21,10,22,5 X_17,14,18,15 X₂₅₃₆ X_13,4,14,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 7, -4, 8}, {-5, 2, -11, 9, -7, 3, -9, 5, -6, 4, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+t(3)t(2)^{2}-t(2)^{2}-3t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+3t(1)t(3)t(2)-3t(3)t(2)+3t(2)+t(1)t(3)^{2}-t(3)^{2}-t(1)t(3)+3t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$1-2q^{-1}+6q^{-2}-7q^{-3}+10q^{-4}-10q^{-5}+10q^{-6}-7q^{-7}+5q^{-8}-2q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+a^{8}z^{4}+3a^{8}z^{2}+2a^{8}-a^{6}z^{6}-3a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-a^{4}z^{6}-3a^{4}z^{4}-3a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+a^{2}z^{4}+3a^{2}z^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$3a^{11}z^{3}-2a^{11}z+a^{10}z^{6}+4a^{10}z^{4}-5a^{10}z^{2}+2a^{10}+3a^{9}z^{7}-4a^{9}z^{5}+8a^{9}z^{3}-4a^{9}z+3a^{8}z^{8}-5a^{8}z^{6}+11a^{8}z^{4}-12a^{8}z^{2}+4a^{8}+a^{7}z^{9}+4a^{7}z^{7}-9a^{7}z^{5}+4a^{7}z^{3}+5a^{6}z^{8}-8a^{6}z^{6}+a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+a^{5}z^{9}+3a^{5}z^{7}-10a^{5}z^{5}+6a^{5}z-2a^{5}z^{-1}+2a^{4}z^{8}-a^{4}z^{6}-11a^{4}z^{4}+14a^{4}z^{2}+2a^{4}z^{-2}-7a^{4}+2a^{3}z^{7}-5a^{3}z^{5}+a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+a^{2}z^{6}-4a^{2}z^{4}+6a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

5

1

4

-5

3

2

-1

-7

7

4

3

-9

5

0

-11

5

0

-13

3

6

3

-15

2

4

-2

-17

3

-19

2

-2

Integral Khovanov Homology

(db, data source)

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