L10a31

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a31 at Knotilus!
	[edit L10a31 Quick Notes]

[edit L10a31 Further Notes and Views]

Link Presentations

[edit Notes on L10a31's Link Presentations]

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

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	$-q^{9/2}+3q^{7/2}-6q^{5/2}+7q^{3/2}-9{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{5}+a^{5}z^{-1}-2z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}+z^{5}a+2z^{3}a+2za+az^{-1}+z^{5}a^{-1}+2z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}-z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-6a^{2}z^{8}-4z^{8}-a^{5}z^{7}-a^{3}z^{7}-6az^{7}-6z^{7}a^{-1}+9a^{4}z^{6}+22a^{2}z^{6}-7z^{6}a^{-2}+6z^{6}+5a^{5}z^{5}+18a^{3}z^{5}+27az^{5}+8z^{5}a^{-1}-6z^{5}a^{-3}-12a^{4}z^{4}-24a^{2}z^{4}+9z^{4}a^{-2}-3z^{4}a^{-4}-8a^{5}z^{3}-30a^{3}z^{3}-27az^{3}+2z^{3}a^{-1}+6z^{3}a^{-3}-z^{3}a^{-5}+5a^{4}z^{2}+11a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}+5a^{5}z+15a^{3}z+10az-3za^{-1}-3za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-2a^{3}z^{-1}-az^{-1}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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

10

1

8

2

-2

6

4

1

3

4

3

2

-1

2

6

4

2

0

5

0

-2

3

4

-1

-4

3

5

2

-6

1

3

-2

-8

1

3

2

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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