L11a453

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_18,11,19,12 X_16,8,17,7 X_20,17,21,18 X_12,19,5,20 X_8,22,9,21 X_22,10,13,9 X_10,14,11,13 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, 3, -6}, {9, -2, 11, -4, 5, -3, 6, -5, 7, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(vw-v+1)(vw-w+1)(uvw-uw+u-vw+v-1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-12q^{4}+16q^{3}-16q^{2}+18q-14+11q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+6z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-a^{2}z^{4}-15z^{4}a^{-2}-3a^{2}z^{2}-18z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}-9a^{-2}+2z^{6}+9z^{4}+14z^{2}+z^{-2}+8$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-6z^{4}a^{-6}+4z^{2}a^{-6}-a^{-6}+9z^{7}a^{-5}-14z^{5}a^{-5}+11z^{3}a^{-5}-za^{-5}+9z^{8}a^{-4}-15z^{6}a^{-4}+11z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+5z^{9}a^{-3}+a^{3}z^{7}+z^{7}a^{-3}-4a^{3}z^{5}-22z^{5}a^{-3}+5a^{3}z^{3}+22z^{3}a^{-3}-2a^{3}z-8za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+15z^{8}a^{-2}-12a^{2}z^{6}-51z^{6}a^{-2}+16a^{2}z^{4}+58z^{4}a^{-2}-9a^{2}z^{2}-37z^{2}a^{-2}-2a^{-2}z^{-2}+3a^{2}+11a^{-2}+3az^{9}+8z^{9}a^{-1}-6az^{7}-15z^{7}a^{-1}-6az^{5}-7z^{5}a^{-1}+16az^{3}+20z^{3}a^{-1}-7az-12za^{-1}+2a^{-1}z^{-1}+z^{10}+9z^{8}-42z^{6}+56z^{4}-35z^{2}-z^{-2}+11$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

2

-2

11

5

1

4

9

8

3

-5

7

8

4

5

8

0

3

10

8

2

1

7

11

4

-1

4

7

-3

2

7

5

-5

1

4

-3

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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