L11a528

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(t(2)t(3)^{2}t(1)^{2}-t(3)^{2}t(1)^{2}+t(2)t(1)^{2}+t(2)^{2}t(3)t(1)^{2}-t(2)t(3)t(1)^{2}+t(3)t(1)^{2}-t(1)^{2}+t(2)^{2}t(1)+t(2)^{2}t(3)^{2}t(1)-2t(2)t(3)^{2}t(1)+t(3)^{2}t(1)-2t(2)t(1)-t(2)^{2}t(3)t(1)-t(3)t(1)+t(1)-t(2)^{2}-t(2)^{2}t(3)^{2}+t(2)t(3)^{2}+t(2)+t(2)^{2}t(3)-t(2)t(3)+t(3)\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+10q^{3}-13q^{2}+15q-14+14q^{-1}-9q^{-2}+7q^{-3}-3q^{-4}+q^{-5}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+2z^{6}-3a^{2}z^{4}+2z^{4}a^{-2}-z^{4}a^{-4}+8z^{4}+a^{4}z^{2}-9a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+11z^{2}+2a^{4}-7a^{2}-2a^{-2}+7+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+3a^{3}z^{9}+11az^{9}+8z^{9}a^{-1}+a^{4}z^{8}-a^{2}z^{8}+13z^{8}a^{-2}+11z^{8}-14a^{3}z^{7}-44az^{7}-17z^{7}a^{-1}+13z^{7}a^{-3}-5a^{4}z^{6}-28a^{2}z^{6}-36z^{6}a^{-2}+10z^{6}a^{-4}-69z^{6}+20a^{3}z^{5}+46az^{5}-8z^{5}a^{-1}-28z^{5}a^{-3}+6z^{5}a^{-5}+10a^{4}z^{4}+63a^{2}z^{4}+28z^{4}a^{-2}-14z^{4}a^{-4}+3z^{4}a^{-6}+98z^{4}-7a^{3}z^{3}-5az^{3}+20z^{3}a^{-1}+14z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}-10a^{4}z^{2}-46a^{2}z^{2}-14z^{2}a^{-2}+4z^{2}a^{-4}-54z^{2}-3a^{3}z-7az-6za^{-1}-2za^{-3}+5a^{4}+13a^{2}+4a^{-2}+13+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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

13

1

-1

11

2

9

4

1

-3

7

6

2

4

5

7

4

-3

3

8

6

2

1

8

9

1

-1

6

0

-3

5

10

5

-5

2

4

-2

-7

1

5

4

-9

2

-2

-11

1

Integral Khovanov Homology

(db, data source)

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