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

L11a135 L11a137

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a136 Visit L11a136's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X14,3,15,4 X10,5,11,6 X22,11,5,12 X20,13,21,14 X12,21,13,22 X4,19,1,20 X18,16,19,15 X16,8,17,7 X8,18,9,17

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 3q-15/2 - 6q-13/2 + 9q-11/2 - 13q-9/2 + 15q-7/2 - 16q-5/2 + 14q-3/2 - 11q-1/2 + 7q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-26 - q-24 + q-22 + 3q-20 - 2q-18 + 2q-16 + q-14 - 2q-12 + 2q-10 - q-8 + 2q-6 + q-4 - q-2 + 4 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az-1 - 2az - az3 - az5 + 2a3z-1 + 3a3z + a3z3 - a3z5 - 2a5z-1 - 4a5z - 2a5z3 - a5z5 + a7z-1 + a7z + a7z3

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 + 7z4 - 7z6 + az-1 - 3az + 2az3 + 8az5 - 8az7 + 2a2z2 - 7a2z4 + 13a2z6 - 8a2z8 + 2a3z-1 - 8a3z + 11a3z3 - 11a3z5 + 12a3z7 - 6a3z9 + a4 + 6a4z2 - 29a4z4 + 28a4z6 - 5a4z8 - 2a4z10 + 2a5z-1 - 9a5z + 24a5z3 - 43a5z5 + 36a5z7 - 10a5z9 + 9a6z2 - 27a6z4 + 20a6z6 - 2a6z10 + a7z-1 - 3a7z + 8a7z3 - 16a7z5 + 15a7z7 - 4a7z9 + 5a8z2 - 13a8z4 + 12a8z6 - 3a8z8 + a9z - 4a9z3 + 4a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                       1 j = 4                     3   j = 2                   4 1   j = 0                 7 3     j = -2               8 5       j = -4             8 6         j = -6           7 8           j = -8         6 8             j = -10       4 8               j = -12     2 5                 j = -14   1 4                   j = -16   2                     j = -18 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, 136]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 136]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[14, 3, 15, 4], X[10, 5, 11, 6], > X[22, 11, 5, 12], X[20, 13, 21, 14], X[12, 21, 13, 22], X[4, 19, 1, 20], > X[18, 16, 19, 15], X[16, 8, 17, 7], X[8, 18, 9, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, > 8, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 9 13 15 16 14 11 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 7 Sqrt[q] - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 -22 3 2 2 -14 2 2 -8 2 -4 4 + q - q + q + --- - --- + --- + q - --- + --- - q + -- + q - 20 18 16 12 10 6 q q q q q q -2 2 6 8 > q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 136]][a, z]
Out[8]=	3 5 7 3 a 2 a 2 a a 3 5 7 z 3 3 3 -(-) + ---- - ---- + -- - 2 a z + 3 a z - 4 a z + a z + -- - a z + a z - z z z z a 5 3 7 3 5 3 5 5 5 > 2 a z + a z - a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 136]][a, z]
Out[9]=	3 5 7 4 a 2 a 2 a a 3 5 7 9 2 2 a + - + ---- + ---- + -- - 3 a z - 8 a z - 9 a z - 3 a z + a z + 2 a z + z z z z 3 4 2 6 2 8 2 3 z 3 3 3 5 3 > 6 a z + 9 a z + 5 a z + ---- + 2 a z + 11 a z + 24 a z + a 4 7 3 9 3 4 z 2 4 4 4 6 4 8 4 > 8 a z - 4 a z + 7 z - -- - 7 a z - 29 a z - 27 a z - 13 a z - 2 a 5 4 z 5 3 5 5 5 7 5 9 5 6 > ---- + 8 a z - 11 a z - 43 a z - 16 a z + 4 a z - 7 z + a 2 6 4 6 6 6 8 6 7 3 7 5 7 > 13 a z + 28 a z + 20 a z + 12 a z - 8 a z + 12 a z + 36 a z + 7 7 9 7 2 8 4 8 8 8 3 9 5 9 > 15 a z - a z - 8 a z - 5 a z - 3 a z - 6 a z - 10 a z - 7 9 4 10 6 10 > 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 2 5 4 7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 8 6 8 7 8 8 6 8 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t 2 2 2 4 2 6 3 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a136

L11a135 L11a137