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

L11a6 L11a8

Knotscape

This page is passe. Go here instead! The 2-Component Link L11a7 Visit L11a7's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,5,13 X14,22,15,21 X18,10,19,9 X20,12,21,11 X10,20,11,19 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-3/2 - 4q-1/2 + 7q1/2 - 12q3/2 + 14q5/2 - 17q7/2 + 16q9/2 - 14q11/2 + 10q13/2 - 5q15/2 + 3q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + 2q-2 + q2 + 3q4 - q6 + 6q8 + q10 + 3q12 + 2q14 - 3q16 + q18 - 4q20 - q22 - q26 + q28

HOMFLY-PT Polynomial:

a-7z-1 - a-7z - 3a-7z3 - a-7z5 - 2a-5z-1 + a-5z + 5a-5z3 + 4a-5z5 + a-5z7 - a-3z + a-3z3 + 3a-3z5 + a-3z7 + a-1z-1 + a-1z - 2a-1z3 - a-1z5

Kauffman Polynomial:

2a-11z3 - a-11z5 - 4a-10z2 + 7a-10z4 - 3a-10z6 - 3a-9z3 + 7a-9z5 - 4a-9z7 + 2a-8 - 7a-8z2 + 3a-8z4 + 4a-8z6 - 4a-8z8 - a-7z-1 - 2a-7z + 9a-7z3 - 13a-7z5 + 8a-7z7 - 4a-7z9 + 5a-6 - 6a-6z2 - a-6z4 + 2a-6z6 - 2a-6z10 - 2a-5z-1 - 3a-5z + 24a-5z3 - 38a-5z5 + 27a-5z7 - 9a-5z9 + 3a-4 - 2a-4z2 - 8a-4z4 + 12a-4z6 - 2a-4z8 - 2a-4z10 - 2a-3z + 4a-3z3 - 6a-3z5 + 11a-3z7 - 5a-3z9 - a-2 + a-2z2 - 9a-2z4 + 16a-2z6 - 6a-2z8 + a-1z-1 - a-1z - 6a-1z3 + 11a-1z5 - 4a-1z7 + 2z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 20                       1 j = 18                     2   j = 16                   3 1   j = 14                 7 2     j = 12               7 3       j = 10             9 7         j = 8           8 7           j = 6         6 9             j = 4       6 8               j = 2     3 8                 j = 0   1 4                   j = -2   3                     j = -4 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, 7]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 7]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[22, 14, 5, 13], X[14, 22, 15, 21], X[18, 10, 19, 9], > X[20, 12, 21, 11], X[10, 20, 11, 19], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 8, -10, 9, -4, 6, -7, 11, -2, 3, -8, > 10, -9, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 4 3/2 5/2 7/2 9/2 q - ------- + 7 Sqrt[q] - 12 q + 14 q - 17 q + 16 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 14 q + 10 q - 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 2 2 4 6 8 10 12 14 16 18 20 -q + -- + q + 3 q - q + 6 q + q + 3 q + 2 q - 3 q + q - 4 q - 2 q 22 26 28 > q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 7]][a, z]
Out[8]=	3 3 3 3 5 5 1 2 1 z z z z 3 z 5 z z 2 z z 4 z ---- - ---- + --- - -- + -- - -- + - - ---- + ---- + -- - ---- - -- + ---- + 7 5 a z 7 5 3 a 7 5 3 a 7 5 a z a z a a a a a a a a 5 5 7 7 3 z z z z > ---- - -- + -- + -- 3 a 5 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 7]][a, z]
Out[9]=	2 2 2 5 3 -2 1 2 1 2 z 3 z 2 z z 4 z 7 z -- + -- + -- - a - ---- - ---- + --- - --- - --- - --- - - - ---- - ---- - 8 6 4 7 5 a z 7 5 3 a 10 8 a a a a z a z a a a a a 2 2 2 3 3 3 3 3 3 4 6 z 2 z z 2 z 3 z 9 z 24 z 4 z 6 z 4 7 z > ---- - ---- + -- + ---- - ---- + ---- + ----- + ---- - ---- + 2 z + ---- + 6 4 2 11 9 7 5 3 a 10 a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 3 z z 8 z 9 z z 7 z 13 z 38 z 6 z 11 z 6 > ---- - -- - ---- - ---- - --- + ---- - ----- - ----- - ---- + ----- - z - 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 3 z 4 z 2 z 12 z 16 z 4 z 8 z 27 z 11 z 4 z > ---- + ---- + ---- + ----- + ----- - ---- + ---- + ----- + ----- - ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 8 8 8 9 9 9 10 10 4 z 2 z 6 z 4 z 9 z 5 z 2 z 2 z > ---- - ---- - ---- - ---- - ---- - ---- - ----- - ----- 8 4 2 7 5 3 6 4 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 3 4 3 q 4 6 6 2 8 q + 6 q + ----- + t + ----- + - + ---- + 8 q t + 6 q t + 9 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 14 5 > 8 q t + 7 q t + 9 q t + 7 q t + 7 q t + 3 q t + 7 q t + 14 6 16 6 16 7 18 7 20 8 > 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a7

L11a6 L11a8