L11a545

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,5,15,6 X_12,4,13,3 X_2,9,3,10 X_18,7,19,8 X_8,17,9,18 X_10,13,5,14 X_22,20,17,19 X_16,21,11,22 X_20,11,21,12 X_4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -6, 4, -7}, {10, -3, 7, -2, 11, -9}, {6, -5, 8, -10, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{2}x-2uv^{2}w^{2}-2uv^{2}wx+3uv^{2}w+uv^{2}x-uv^{2}-2uvw^{2}x+2uvw^{2}+5uvwx-5uvw-3uvx+2uv-2uwx+2uw+2ux-u-v^{2}w^{2}x+2v^{2}w^{2}+2v^{2}wx-2v^{2}w+2vw^{2}x-3vw^{2}-5vwx+5vw+2vx-2v-w^{2}x+w^{2}+3wx-2w-2x+1}{{\sqrt {u}}vw{\sqrt {x}}}}$ (db)
Jones polynomial	${\frac {21}{q^{9/2}}}-{\frac {25}{q^{7/2}}}+{\frac {20}{q^{5/2}}}+q^{3/2}-{\frac {16}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {8}{q^{15/2}}}+{\frac {13}{q^{13/2}}}-{\frac {21}{q^{11/2}}}-5{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}+a^{9}z^{-3}-3z^{3}a^{7}-6za^{7}-6a^{7}z^{-1}-3a^{7}z^{-3}+3z^{5}a^{5}+8z^{3}a^{5}+10za^{5}+9a^{5}z^{-1}+3a^{5}z^{-3}-z^{7}a^{3}-3z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}+z^{5}a+z^{3}a-za$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-10a^{9}z^{5}+12a^{9}z^{3}-a^{9}z^{-3}-11a^{9}z+5a^{9}z^{-1}+7a^{8}z^{8}-6a^{8}z^{6}-4a^{8}z^{4}+14a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+5a^{7}z^{9}+7a^{7}z^{7}-35a^{7}z^{5}+50a^{7}z^{3}-3a^{7}z^{-3}-33a^{7}z+12a^{7}z^{-1}+2a^{6}z^{10}+13a^{6}z^{8}-25a^{6}z^{6}+26a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+12a^{5}z^{9}-10a^{5}z^{7}-27a^{5}z^{5}+46a^{5}z^{3}-3a^{5}z^{-3}-33a^{5}z+12a^{5}z^{-1}+2a^{4}z^{10}+15a^{4}z^{8}-37a^{4}z^{6}+11a^{4}z^{4}+14a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+7a^{3}z^{9}-6a^{3}z^{7}-13a^{3}z^{5}+14a^{3}z^{3}-a^{3}z^{-3}-11a^{3}z+5a^{3}z^{-1}+9a^{2}z^{8}-20a^{2}z^{6}+10a^{2}z^{4}+a^{2}z^{2}+5az^{7}-10az^{5}+4az^{3}+az+z^{6}-z^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

4

0

6

1

-5

-2

10

4

6

-4

12

8

-4

-6

13

8

5

-8

10

14

4

-10

11

0

-12

4

12

8

-14

4

9

-5

-16

1

6

5

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$