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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X10,6,11,5 X8493 X14,10,5,9 X2,12,3,11

Gauss Code:

{{1, -7, 5, -3}, {4, -1, 2, -5, 6, -4, 7, -2, 3, -6}}

Jones Polynomial:

q-5/2 - 3q-3/2 + 3q-1/2 - 5q1/2 + 4q3/2 - 4q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-8 + q-6 + q-4 + 2q-2 + 4 + q2 + 2q4 - q6 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z - a-3z3 - a-1z-1 + 2a-1z + 3a-1z3 + a-1z5 + az-1 - az - az3

Kauffman Polynomial:

- a-5z3 + 2a-4z2 - 3a-4z4 - 2a-3z + 5a-3z3 - 4a-3z5 + 3a-2z2 - a-2z4 - 2a-2z6 - a-1z-1 - 4a-1z + 12a-1z3 - 7a-1z5 + 1 + 2z2 + z4 - 2z6 - az-1 - 2az + 6az3 - 3az5 + a2z2 - a2z4

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L7a1

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