L10a154

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a154 at Knotilus!
	[edit L10a154 Quick Notes]

[edit L10a154 Further Notes and Views]

Link Presentations

[edit Notes on L10a154's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(2uvw-3uv+uw^{2}-3uw+u+vw^{2}-3vw+v-3w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$q-3+7q^{-1}-11q^{-2}+13q^{-3}-13q^{-4}+13q^{-5}-9q^{-6}+7q^{-7}-2q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-2a^{8}z^{-2}-4a^{8}+6a^{6}z^{2}+a^{6}z^{-2}+6a^{6}-3a^{4}z^{4}-5a^{4}z^{2}-4a^{4}-a^{2}z^{4}+2a^{2}z^{2}+2a^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{10}-4z^{4}a^{10}+6z^{2}a^{10}+a^{10}z^{-2}-4a^{10}+2z^{7}a^{9}-4z^{5}a^{9}+4za^{9}-2a^{9}z^{-1}+3z^{8}a^{8}-5z^{6}a^{8}-z^{4}a^{8}+5z^{2}a^{8}+2a^{8}z^{-2}-5a^{8}+z^{9}a^{7}+9z^{7}a^{7}-29z^{5}a^{7}+22z^{3}a^{7}-4za^{7}-2a^{7}z^{-1}+8z^{8}a^{6}-11z^{6}a^{6}-5z^{4}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}-2a^{6}+z^{9}a^{5}+15z^{7}a^{5}-41z^{5}a^{5}+36z^{3}a^{5}-12za^{5}+5z^{8}a^{4}+z^{6}a^{4}-16z^{4}a^{4}+14z^{2}a^{4}-2a^{4}+8z^{7}a^{3}-13z^{5}a^{3}+12z^{3}a^{3}-4za^{3}+6z^{6}a^{2}-7z^{4}a^{2}+6z^{2}a^{2}-2a^{2}+3z^{5}a-2z^{3}a+z^{4}-z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

2

-2

-1

5

1

4

-3

7

3

-4

-5

6

4

2

-7

7

0

-9

6

0

-11

5

9

4

-13

2

4

-2

-15

5

-17

1

2

-1

-19

1

Integral Khovanov Homology

(db, data source)

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