© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11n152 L11n154

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n153 Visit L11n153's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X7,14,8,15 X13,20,14,21 X19,1,20,6 X18,11,19,12 X5,12,6,13 X15,22,16,7 X4,18,5,17 X21,16,22,17

Gauss Code:

{{1, -2, 3, -10, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, -9, 11, 10, -7, -6, 5, -11, 9}}

Jones Polynomial:

q-21/2 - 2q-19/2 + 4q-17/2 - 6q-15/2 + 7q-13/2 - 8q-11/2 + 7q-9/2 - 6q-7/2 + 3q-5/2 - 2q-3/2

A2 (sl(3)) Invariant:

- q-32 - q-28 - 2q-26 + q-24 - q-22 + q-20 + 2q-18 + q-16 + 3q-14 + 2q-10 + 2q-8 + 2q-4

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 2a3z3 + a5z + 2a5z3 + a5z5 + 2a7z-1 + 4a7z + 3a7z3 + a7z5 - a9z-1 - 2a9z - a9z3

Kauffman Polynomial:

- a3z-1 + 5a3z - 3a3z3 + a4 - a4z2 - a4z6 + 2a5z - 7a5z3 + 4a5z5 - 2a5z7 - 3a6 + 7a6z2 - 10a6z4 + 5a6z6 - 2a6z8 + 2a7z-1 - 7a7z + 7a7z3 - 3a7z5 + a7z7 - a7z9 - 5a8 + 18a8z2 - 20a8z4 + 13a8z6 - 4a8z8 + a9z-1 - 3a9z + 5a9z3 + a9z7 - a9z9 - 2a10 + 6a10z2 - 6a10z4 + 6a10z6 - 2a10z8 + a11z - 6a11z3 + 7a11z5 - 2a11z7 - 4a12z2 + 4a12z4 - a12z6

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -2                   2 j = -4                 2 1 j = -6               4 1   j = -8             4 3     j = -10           4 3       j = -12         3 4         j = -14       3 4           j = -16     1 3             j = -18   1 3               j = -20   1                 j = -22 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 153]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 153]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[7, 14, 8, 15], > X[13, 20, 14, 21], X[19, 1, 20, 6], X[18, 11, 19, 12], X[5, 12, 6, 13], > X[15, 22, 16, 7], X[4, 18, 5, 17], X[21, 16, 22, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, -8, 6}, > {-4, -1, 2, -3, 7, 8, -5, 4, -9, 11, 10, -7, -6, 5, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(21/2) 2 4 6 7 8 7 6 3 2 q - ----- + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -28 2 -24 -22 -20 2 -16 3 2 2 2 -q - q - --- + q - q + q + --- + q + --- + --- + -- + -- 26 18 14 10 8 4 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 153]][a, z]
Out[8]=	3 7 9 a 2 a a 3 5 7 9 3 3 5 3 -(--) + ---- - -- - 4 a z + a z + 4 a z - 2 a z - 2 a z + 2 a z + z z z 7 3 9 3 5 5 7 5 > 3 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 153]][a, z]
Out[9]=	3 7 9 4 6 8 10 a 2 a a 3 5 7 9 a - 3 a - 5 a - 2 a - -- + ---- + -- + 5 a z + 2 a z - 7 a z - 3 a z + z z z 11 4 2 6 2 8 2 10 2 12 2 3 3 > a z - a z + 7 a z + 18 a z + 6 a z - 4 a z - 3 a z - 5 3 7 3 9 3 11 3 6 4 8 4 10 4 > 7 a z + 7 a z + 5 a z - 6 a z - 10 a z - 20 a z - 6 a z + 12 4 5 5 7 5 11 5 4 6 6 6 8 6 > 4 a z + 4 a z - 3 a z + 7 a z - a z + 5 a z + 13 a z + 10 6 12 6 5 7 7 7 9 7 11 7 6 8 > 6 a z - a z - 2 a z + a z + a z - 2 a z - 2 a z - 8 8 10 8 7 9 9 9 > 4 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-4 2 1 1 1 3 1 3 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 22 9 20 8 18 8 18 7 16 7 16 6 14 6 q q t q t q t q t q t q t q t 4 3 4 4 3 4 3 4 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 14 5 12 5 12 4 10 4 10 3 8 3 8 2 6 2 6 q t q t q t q t q t q t q t q t q t 2 > ---- 4 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n153

L11n152 L11n154