L10a136

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a136 at Knotilus!
	[edit L10a136 Quick Notes]

[edit L10a136 Further Notes and Views]

Link Presentations

[edit Notes on L10a136's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)t(2)-t(2)-2t(3)+1)(t(3)t(2)-2t(2)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{5}-q^{-5}+4q^{4}+4q^{-4}-8q^{3}-8q^{-3}+13q^{2}+13q^{-2}-15q-15q^{-1}+18$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}-z^{2}a^{-4}+2a^{2}z^{4}+2z^{4}a^{-2}+2a^{2}z^{2}+2z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{2}+a^{-2}-z^{6}-2z^{4}-3z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+6a^{2}z^{8}+6z^{8}a^{-2}+12z^{8}+7a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+7z^{7}a^{-3}+4a^{4}z^{6}-3a^{2}z^{6}-3z^{6}a^{-2}+4z^{6}a^{-4}-14z^{6}+a^{5}z^{5}-11a^{3}z^{5}-28az^{5}-28z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-7a^{2}z^{4}-7z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+12z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+3a^{4}z^{2}+5a^{2}z^{2}+5z^{2}a^{-2}+3z^{2}a^{-4}+4z^{2}+2az+2za^{-1}-2a^{2}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

11

1

-1

9

3

7

5

1

-4

5

8

3

5

3

8

6

-2

1

10

7

3

-1

7

10

3

-3

6

8

-2

-5

3

8

5

-7

1

5

-4

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

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