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

L11a339 L11a341

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a340 Visit L11a340's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X10,1,11,2 X14,4,15,3 X22,5,9,6 X6,9,7,10 X20,12,21,11 X18,14,19,13 X12,20,13,19 X16,8,17,7 X4,16,5,15 X8,18,1,17 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 8q-3/2 + 13q-1/2 - 17q1/2 + 19q3/2 - 20q5/2 + 16q7/2 - 12q9/2 + 7q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - q-8 + 3q-6 + q-4 - 2q-2 + 4 - 3q2 + 3q4 + q6 + 4q10 - 3q12 + 3q14 - 2q18 + q20 - q22

HOMFLY-PT Polynomial:

3a-5z + 3a-5z3 + a-5z5 - 2a-3z - 6a-3z3 - 4a-3z5 - a-3z7 - a-1z-1 - 5a-1z - 7a-1z3 - 4a-1z5 - a-1z7 + az-1 + 4az + 3az3 + az5

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + a-7z - 5a-7z3 + 8a-7z5 - 3a-7z7 + 5a-6z2 - 11a-6z4 + 13a-6z6 - 5a-6z8 - 3a-5z + 8a-5z3 - 10a-5z5 + 11a-5z7 - 5a-5z9 + 7a-4z2 - 20a-4z4 + 20a-4z6 - 5a-4z8 - 2a-4z10 - 2a-3z + 6a-3z3 - 17a-3z5 + 21a-3z7 - 10a-3z9 + 4a-2z2 - 16a-2z4 + 19a-2z6 - 7a-2z8 - 2a-2z10 - a-1z-1 + 8a-1z - 17a-1z3 + 12a-1z5 + a-1z7 - 5a-1z9 + 1 + 3z2 - 6z4 + 10z6 - 7z8 - az-1 + 5az - 8az3 + 10az5 - 6az7 - a2z2 + 4a2z4 - 3a2z6 - a3z + 2a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                     2   j = 12                   5 1   j = 10                 7 2     j = 8               9 5       j = 6             11 7         j = 4           8 9           j = 2         9 11             j = 0       6 10               j = -2     2 7                 j = -4   1 6                   j = -6   2                     j = -8 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, 340]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 340]]
Out[4]=	PD[X[10, 1, 11, 2], X[14, 4, 15, 3], X[22, 5, 9, 6], X[6, 9, 7, 10], > X[20, 12, 21, 11], X[18, 14, 19, 13], X[12, 20, 13, 19], X[16, 8, 17, 7], > X[4, 16, 5, 15], X[8, 18, 1, 17], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 3, -4, 8, -10}, > {4, -1, 5, -7, 6, -2, 9, -8, 10, -6, 7, -5, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 8 13 3/2 5/2 7/2 -q + ---- - ---- + ------- - 17 Sqrt[q] + 19 q - 20 q + 16 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 12 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 3 -4 2 2 4 6 10 12 14 4 + q - q + -- + q - -- - 3 q + 3 q + q + 4 q - 3 q + 3 q - 6 2 q q 18 20 22 > 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 340]][a, z]
Out[8]=	3 3 3 5 1 a 3 z 2 z 5 z 3 z 6 z 7 z 3 z -(---) + - + --- - --- - --- + 4 a z + ---- - ---- - ---- + 3 a z + -- - a z z 5 3 a 5 3 a 5 a a a a a 5 5 7 7 4 z 4 z 5 z z > ---- - ---- + a z - -- - -- 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 340]][a, z]
Out[9]=	2 2 2 1 a z 3 z 2 z 8 z 3 2 2 z 5 z 7 z 1 - --- - - + -- - --- - --- + --- + 5 a z - a z + 3 z - ---- + ---- + ---- + a z z 7 5 3 a 8 6 4 a a a a a a 2 3 3 3 3 4 z 2 2 5 z 8 z 6 z 17 z 3 3 3 4 > ---- - a z - ---- + ---- + ---- - ----- - 8 a z + 2 a z - 6 z + 2 7 5 3 a a a a a 4 4 4 4 5 5 5 5 3 z 11 z 20 z 16 z 2 4 8 z 10 z 17 z 12 z > ---- - ----- - ----- - ----- + 4 a z + ---- - ----- - ----- + ----- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 6 7 5 3 5 6 z 13 z 20 z 19 z 2 6 3 z > 10 a z - a z + 10 z - -- + ----- + ----- + ----- - 3 a z - ---- + 8 6 4 2 7 a a a a a 7 7 7 8 8 8 9 9 11 z 21 z z 7 8 5 z 5 z 7 z 5 z 10 z > ----- + ----- + -- - 6 a z - 7 z - ---- - ---- - ---- - ---- - ----- - 5 3 a 6 4 2 5 3 a a a a a a a 9 10 10 5 z 2 z 2 z > ---- - ----- - ----- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 6 2 6 7 2 10 + 9 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 11 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 8 q t + 9 q t + 11 q t + 7 q t + 9 q t + 5 q t + 7 q t + 10 5 12 5 12 6 14 6 16 7 > 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a340

L11a339 L11a341