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

L11n172 L11n174

Knotscape

This page is passe. Go here instead! The 2-Component Link L11n173 Visit L11n173's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X12,7,13,8 X15,7,16,22 X14,6,15,5 X6,14,1,13 X21,17,22,16 X18,10,19,9 X20,11,21,12 X4,18,5,17 X2,19,3,20

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 8q-3/2 + 11q-1/2 - 14q1/2 + 14q3/2 - 13q5/2 + 9q7/2 - 6q9/2 + 2q11/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 2q-6 - q-4 - q-2 + 4 - 2q2 + 4q4 - q6 + 2q8 + 3q10 - q12 + 3q14 - q16 - q18

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

3a-6z2 - 3a-6z4 - 3a-5z + 9a-5z3 - 5a-5z5 - a-5z7 + 4a-4z2 - 5a-4z4 + 3a-4z6 - 3a-4z8 + a-3z-1 - 6a-3z + 9a-3z3 - 2a-3z5 - a-3z7 - 2a-3z9 - a-2 + 2a-2z2 - 9a-2z4 + 15a-2z6 - 9a-2z8 + a-1z-1 - 2a-1z - 7a-1z3 + 17a-1z5 - 7a-1z7 - 2a-1z9 - z4 + 8z6 - 6z8 + az - 6az3 + 13az5 - 7az7 - a2z2 + 6a2z4 - 4a2z6 + a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                   2 j = 10                 4   j = 8               5 2   j = 6             8 4     j = 4           7 6       j = 2         7 7         j = 0       5 8           j = -2     3 6             j = -4   1 5               j = -6   3                 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, NonAlternating, 173]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 173]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[12, 7, 13, 8], X[15, 7, 16, 22], > X[14, 6, 15, 5], X[6, 14, 1, 13], X[21, 17, 22, 16], X[18, 10, 19, 9], > X[20, 11, 21, 12], X[4, 18, 5, 17], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 5, -6}, > {3, -1, 8, -2, 9, -3, 6, -5, -4, 7, 10, -8, 11, -9, -7, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 8 11 3/2 5/2 7/2 -q + ---- - ---- + ------- - 14 Sqrt[q] + 14 q - 13 q + 9 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 6 q + 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 -4 -2 2 4 6 8 10 12 4 + q - -- + -- - q - q - 2 q + 4 q - q + 2 q + 3 q - q + 8 6 q q 14 16 18 > 3 q - q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 173]][a, z]
Out[8]=	3 3 5 5 7 1 1 z z z 5 z 3 z 4 z 5 z -(----) + --- + -- - -- + a z + -- - ---- + 2 a z + -- - ---- + a z - -- 3 a z 5 3 3 a 3 a a a z a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 173]][a, z]
Out[9]=	2 2 2 3 -2 1 1 3 z 6 z 2 z 3 z 4 z 2 z 2 2 9 z -a + ---- + --- - --- - --- - --- + a z + ---- + ---- + ---- - a z + ---- + 3 a z 5 3 a 6 4 2 5 a z a a a a a a 3 3 4 4 4 5 9 z 7 z 3 3 3 4 3 z 5 z 9 z 2 4 5 z > ---- - ---- - 6 a z + a z - z - ---- - ---- - ---- + 6 a z - ---- - 3 a 6 4 2 5 a a a a a 5 5 6 6 7 7 2 z 17 z 5 3 5 6 3 z 15 z 2 6 z z > ---- + ----- + 13 a z - a z + 8 z + ---- + ----- - 4 a z - -- - -- - 3 a 4 2 5 3 a a a a a 7 8 8 9 9 7 z 7 8 3 z 9 z 2 z 2 z > ---- - 7 a z - 6 z - ---- - ---- - ---- - ---- a 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 5 6 2 4 8 + 7 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 7 q t + 7 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 2 6 2 6 3 8 3 8 4 10 4 12 5 > 6 q t + 8 q t + 4 q t + 5 q t + 2 q t + 4 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n173

L11n172 L11n174