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

L11a121 L11a123

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a122 Visit L11a122's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X16,12,17,11 X12,16,13,15 X10,18,11,17 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 5q-9/2 + 8q-7/2 - 12q-5/2 + 14q-3/2 - 15q-1/2 + 13q1/2 - 10q3/2 + 7q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + q-16 + 2q-14 - 3q-12 + q-10 + 2q-8 - q-6 + 3q-4 - q-2 + 2 - 2q4 + 3q6 - 2q8 + 2q12 - q14

HOMFLY-PT Polynomial:

a-3z3 - a-1z - a-1z3 - a-1z5 - az-1 - az - az5 + 2a3z-1 + 3a3z + 3a3z3 - 2a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

2a-4z4 - a-4z6 - 5a-3z3 + 11a-3z5 - 4a-3z7 + 2a-2z2 - 13a-2z4 + 18a-2z6 - 6a-2z8 + a-1z - 6a-1z3 + 3a-1z5 + 7a-1z7 - 4a-1z9 + 7z2 - 26z4 + 26z6 - 7z8 - z10 + az-1 - 3az + 6az3 - 14az5 + 14az7 - 6az9 + 3a2z2 - 8a2z4 + 7a2z6 - 3a2z8 - a2z10 + 2a3z-1 - 9a3z + 13a3z3 - 6a3z5 + a3z7 - 2a3z9 + a4 - 4a4z2 + 7a4z4 - 2a4z6 - 2a4z8 + 2a5z-1 - 8a5z + 9a5z3 - a5z5 - 2a5z7 - 2a6z2 + 4a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                       1 j = 8                     3   j = 6                   4 1   j = 4                 6 3     j = 2               7 4       j = 0             8 6         j = -2           7 8           j = -4         5 7             j = -6       3 7               j = -8     2 5                 j = -10   1 4                   j = -12   1                     j = -14 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, Alternating, 122]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 122]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[18, 8, 19, 7], X[22, 20, 5, 19], > X[20, 9, 21, 10], X[8, 21, 9, 22], X[16, 12, 17, 11], X[12, 16, 13, 15], > X[10, 18, 11, 17], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -8, 11, -2, 8, -7, 9, -3, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 5 8 12 14 15 -q + ----- - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 3/2 5/2 7/2 9/2 > 10 q + 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -16 2 3 -10 2 -6 3 -2 4 6 2 + q + --- + q + --- - --- + q + -- - q + -- - q - 2 q + 3 q - 20 14 12 8 4 q q q q q 8 12 14 > 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 122]][a, z]
Out[8]=	3 5 7 3 3 5 a 2 a 2 a a z 3 5 z z 3 3 z -(-) + ---- - ---- + -- - - - a z + 3 a z - 3 a z + -- - -- + 3 a z - -- - z z z z a 3 a a a 5 > a z
In[9]:=	Kauffman[Link[11, Alternating, 122]][a, z]
Out[9]=	3 5 7 4 a 2 a 2 a a z 3 5 7 2 a + - + ---- + ---- + -- + - - 3 a z - 9 a z - 8 a z - 3 a z + 7 z + z z z z a 2 3 3 2 z 2 2 4 2 6 2 5 z 6 z 3 3 3 > ---- + 3 a z - 4 a z - 2 a z - ---- - ---- + 6 a z + 13 a z + 2 3 a a a 4 4 5 3 7 3 4 2 z 13 z 2 4 4 4 6 4 > 9 a z + 3 a z - 26 z + ---- - ----- - 8 a z + 7 a z + 4 a z + 4 2 a a 5 5 6 6 11 z 3 z 5 3 5 5 5 7 5 6 z 18 z > ----- + ---- - 14 a z - 6 a z - a z - a z + 26 z - -- + ----- + 3 a 4 2 a a a 7 7 2 6 4 6 6 6 4 z 7 z 7 3 7 5 7 > 7 a z - 2 a z - 2 a z - ---- + ---- + 14 a z + a z - 2 a z - 3 a a 8 9 8 6 z 2 8 4 8 4 z 9 3 9 10 2 10 > 7 z - ---- - 3 a z - 2 a z - ---- - 6 a z - 2 a z - z - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 1 1 4 2 5 3 7 8 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 5 7 7 2 2 2 4 2 4 3 > ----- + ---- + ---- + 6 t + 7 q t + 4 q t + 6 q t + 3 q t + 4 2 4 2 q t q t q t 6 3 6 4 8 4 10 5 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a122

L11a121 L11a123