L10a125

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{3}-2uvw^{2}+2uvw-2uv-2uw^{3}+3uw^{2}-3uw+2u-2vw^{3}+3vw^{2}-3vw+2v+2w^{3}-2w^{2}+2w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+7q^{2}-10q+11-11q^{-1}+11q^{-2}-6q^{-3}+5q^{-4}-q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-5a^{4}z^{-2}-8a^{4}+3z^{4}a^{2}+9z^{2}a^{2}+4a^{2}z^{-2}+10a^{2}-z^{6}-3z^{4}-4z^{2}-z^{-2}-3+z^{4}a^{-2}+z^{2}a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+9a^{6}z^{2}+2a^{6}z^{-2}-7a^{6}+a^{5}z^{7}-a^{5}z^{5}-6a^{5}z^{3}+11a^{5}z-5a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-14a^{4}z^{4}+z^{4}a^{-4}+20a^{4}z^{2}+5a^{4}z^{-2}-14a^{4}+a^{3}z^{9}+3a^{3}z^{5}+4z^{5}a^{-3}-17a^{3}z^{3}-3z^{3}a^{-3}+21a^{3}z-9a^{3}z^{-1}+5a^{2}z^{8}-7a^{2}z^{6}+7z^{6}a^{-2}-3a^{2}z^{4}-8z^{4}a^{-2}+11a^{2}z^{2}+z^{2}a^{-2}+4a^{2}z^{-2}-10a^{2}+az^{9}+6az^{7}+7z^{7}a^{-1}-8az^{5}-8z^{5}a^{-1}-8az^{3}+13az+3za^{-1}-5az^{-1}-a^{-1}z^{-1}+4z^{8}-z^{6}-3z^{4}+z^{2}+z^{-2}-2$ (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

χ

9

1

7

3

-3

5

4

1

3

6

3

-3

1

5

4

1

-1

7

0

-3

4

0

-5

2

7

5

-7

3

4

-1

-9

1

5

4

-11

0

-13

1

Integral Khovanov Homology

(db, data source)

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