L11n124

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n124 at Knotilus!
	[edit L11n124 Quick Notes]

[edit L11n124 Further Notes and Views]

Link Presentations

[edit Notes on L11n124's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_3,15,4,14 X_9,22,10,5 X_7,19,8,18 X_17,9,18,8 X_19,13,20,12 X_11,21,12,20 X_15,10,16,11 X_21,16,22,17 X₂₅₃₆ X_13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 5, -3, 8, -7, 6, -11, 2, -8, 9, -5, 4, -6, 7, -9, 3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(2v^{2}-3v+2\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$3q^{9/2}-6q^{7/2}+{\frac {1}{q^{7/2}}}+8q^{5/2}-{\frac {4}{q^{5/2}}}-9q^{3/2}+{\frac {6}{q^{3/2}}}-q^{11/2}+9{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-1}+3az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-a^{3}z+5az-9za^{-1}+6za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-10z^{8}a^{-2}-3z^{8}a^{-4}-7z^{8}-7az^{7}-5z^{7}a^{-1}+z^{7}a^{-3}-z^{7}a^{-5}-2a^{2}z^{6}+39z^{6}a^{-2}+12z^{6}a^{-4}+25z^{6}+25az^{5}+42z^{5}a^{-1}+21z^{5}a^{-3}+4z^{5}a^{-5}+a^{2}z^{4}-42z^{4}a^{-2}-14z^{4}a^{-4}-27z^{4}-4a^{3}z^{3}-32az^{3}-56z^{3}a^{-1}-34z^{3}a^{-3}-6z^{3}a^{-5}-a^{4}z^{2}+17z^{2}a^{-2}+6z^{2}a^{-4}+12z^{2}+3a^{3}z+15az+26za^{-1}+18za^{-3}+4za^{-5}-a^{2}-3a^{-2}-a^{-4}-2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

2

-2

8

4

1

3

6

4

2

-2

4

5

4

1

2

1

5

4

0

5

0

-2

3

6

3

-4

1

3

-2

-6

3

-8

1

-1

Integral Khovanov Homology

(db, data source)

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