L10a41

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {2}{q^{9/2}}}-3q^{7/2}+{\frac {4}{q^{7/2}}}+5q^{5/2}-{\frac {7}{q^{5/2}}}-7q^{3/2}+{\frac {8}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+8{\sqrt {q}}-{\frac {10}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{5}-a^{5}z^{-1}+2z^{3}a^{3}+4za^{3}+3a^{3}z^{-1}-z^{5}a-2z^{3}a-3za-2az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{5}-3a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}+2a^{4}z^{6}+z^{6}a^{-4}+a^{4}z^{4}-3z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}+3a^{4}+2a^{3}z^{7}+3z^{7}a^{-3}+2a^{3}z^{5}-10z^{5}a^{-3}-7a^{3}z^{3}+8z^{3}a^{-3}+7a^{3}z-2za^{-3}-3a^{3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}-a^{2}z^{6}-8z^{6}a^{-2}+2a^{2}z^{4}+3z^{4}a^{-2}-4a^{2}z^{2}+z^{2}a^{-2}+3a^{2}+az^{9}+z^{9}a^{-1}+2az^{7}+3z^{7}a^{-1}-5az^{5}-15z^{5}a^{-1}+12z^{3}a^{-1}+5az-2za^{-1}-2az^{-1}+5z^{8}-12z^{6}+8z^{4}-2z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

3

1

-2

4

2

4

3

-1

0

6

4

2

-2

4

6

2

-4

3

4

-1

-6

1

4

3

-8

1

3

-2

-10

1

-12

1

-1

Integral Khovanov Homology

(db, data source)

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