L10a121

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a121 at Knotilus!
	[edit L10a121 Quick Notes]

[edit L10a121 Further Notes and Views]

Link Presentations

[edit Notes on L10a121's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_20,5,11,6 X_16,3,17,4 X_10,15,1,16 X_8,19,9,20 X_18,9,19,10 X_4,17,5,18 X_2,11,3,12 X_6,13,7,14 X_14,7,15,8
Gauss code	{1, -8, 3, -7, 2, -9, 10, -5, 6, -4}, {8, -1, 9, -10, 4, -3, 7, -6, 5, -2}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}\left(-v^{2}\right)-3u^{3}v^{3}+5u^{3}v^{2}-3u^{3}v-u^{2}v^{4}+5u^{2}v^{3}-9u^{2}v^{2}+5u^{2}v-u^{2}-3uv^{3}+5uv^{2}-3uv-v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {14}{q^{13/2}}}+{\frac {16}{q^{15/2}}}-{\frac {14}{q^{17/2}}}+{\frac {11}{q^{19/2}}}-{\frac {8}{q^{21/2}}}+{\frac {3}{q^{23/2}}}-{\frac {1}{q^{25/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}z^{3}+a^{11}z+a^{11}z^{-1}-a^{9}z^{5}+2a^{9}z-a^{9}z^{-1}-3a^{7}z^{5}-9a^{7}z^{3}-7a^{7}z-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$-z^{5}a^{15}+2z^{3}a^{15}-za^{15}-3z^{6}a^{14}+4z^{4}a^{14}-z^{2}a^{14}-6z^{7}a^{13}+12z^{5}a^{13}-12z^{3}a^{13}+7za^{13}-5z^{8}a^{12}+3z^{6}a^{12}+6z^{4}a^{12}-5z^{2}a^{12}-2z^{9}a^{11}-8z^{7}a^{11}+18z^{5}a^{11}-9z^{3}a^{11}+za^{11}+a^{11}z^{-1}-10z^{8}a^{10}+12z^{6}a^{10}+4z^{4}a^{10}-8z^{2}a^{10}-a^{10}-2z^{9}a^{9}-8z^{7}a^{9}+18z^{5}a^{9}-9z^{3}a^{9}+za^{9}+a^{9}z^{-1}-5z^{8}a^{8}+3z^{6}a^{8}+6z^{4}a^{8}-5z^{2}a^{8}-6z^{7}a^{7}+12z^{5}a^{7}-12z^{3}a^{7}+7za^{7}-3z^{6}a^{6}+4z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+2z^{3}a^{5}-za^{5}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

5

-10

6

3

-3

-12

8

5

3

-14

8

6

-2

-16

6

8

-2

-18

5

8

3

-20

3

6

-3

-22

5

-24

1

3

-2

-26

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-10$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{3}$
$r=-8$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-7$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-6$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$