L11a543

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{2}wx-uv^{2}w-uv^{2}x+uv^{2}+2uvw^{2}x-uvw^{2}-5uvwx+4uvw+2uvx-3uv-uw^{2}x+uw^{2}+2uwx-3uw-ux+2u+2v^{2}w^{2}x+v^{2}\left(-w^{2}\right)-3v^{2}wx+2v^{2}w+v^{2}x-v^{2}-3vw^{2}x+2vw^{2}+4vwx-5vw-vx+2v+w^{2}x-w^{2}-wx+2w}{{\sqrt {u}}vw{\sqrt {x}}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{27/2}}}-{\frac {4}{q^{25/2}}}+{\frac {8}{q^{23/2}}}-{\frac {14}{q^{21/2}}}+{\frac {17}{q^{19/2}}}-{\frac {22}{q^{17/2}}}+{\frac {19}{q^{15/2}}}-{\frac {19}{q^{13/2}}}+{\frac {12}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-3}+a^{13}(-z)+3a^{11}z^{3}-3a^{11}z^{-3}+2a^{11}z-5a^{11}z^{-1}-2a^{9}z^{5}-a^{9}z^{3}+3a^{9}z^{-3}+8a^{9}z+10a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-a^{7}z^{-3}-8a^{7}z-5a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$a^{16}z^{6}-2a^{16}z^{4}+a^{16}z^{2}+4a^{15}z^{7}-10a^{15}z^{5}+8a^{15}z^{3}-3a^{15}z+6a^{14}z^{8}-12a^{14}z^{6}+4a^{14}z^{4}+a^{14}z^{2}+4a^{13}z^{9}+4a^{13}z^{7}-31a^{13}z^{5}+33a^{13}z^{3}-a^{13}z^{-3}-15a^{13}z+5a^{13}z^{-1}+a^{12}z^{10}+15a^{12}z^{8}-38a^{12}z^{6}+23a^{12}z^{4}+4a^{12}z^{2}+3a^{12}z^{-2}-10a^{12}+8a^{11}z^{9}+a^{11}z^{7}-38a^{11}z^{5}+49a^{11}z^{3}-3a^{11}z^{-3}-29a^{11}z+12a^{11}z^{-1}+a^{10}z^{10}+15a^{10}z^{8}-31a^{10}z^{6}+12a^{10}z^{4}+18a^{10}z^{2}+6a^{10}z^{-2}-19a^{10}+4a^{9}z^{9}+7a^{9}z^{7}-29a^{9}z^{5}+39a^{9}z^{3}-3a^{9}z^{-3}-29a^{9}z+12a^{9}z^{-1}+6a^{8}z^{8}-3a^{8}z^{6}-9a^{8}z^{4}+15a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+6a^{7}z^{7}-11a^{7}z^{5}+13a^{7}z^{3}-a^{7}z^{-3}-11a^{7}z+5a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

5

-10

7

3

-4

-12

12

5

7

-14

9

0

-16

13

10

3

-18

8

13

5

-20

6

9

-3

-22

3

9

6

-24

1

5

-4

-26

3

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-11$	${\mathbb {Z} }$
$r=-10$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-8$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-7$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-6$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{13}$
$r=-5$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$