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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X9,18,10,15 X8493 X5,17,6,16 X17,5,18,14 X15,10,16,11 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-5 + q-4 + q-2 + q-1 + 1 + q - q2 + q3

A2 (sl(3)) Invariant:

- q-16 - q-14 - q-12 + q-10 + 3q-8 + 4q-6 + 5q-4 + 4q-2 + 4 + 3q2 + 2q4 + 2q6 + q8 + q10

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 - 2z-2 - 6 - 5z2 - z4 + a2z-2 + 6a2 + 5a2z2 + a2z4 - 2a4 - a4z2

Kauffman Polynomial:

a-2z-2 - 4a-2 + 6a-2z2 - 5a-2z4 + a-2z6 - 2a-1z-1 + 2a-1z + 4a-1z3 - 5a-1z5 + a-1z7 + 2z-2 - 11 + 22z2 - 13z4 + 2z6 - 2az-1 + 6az + 2az3 - 5az5 + az7 + a2z-2 - 12a2 + 22a2z2 - 13a2z4 + 2a2z6 + 6a3z - 6a3z3 + a3z5 - 4a4 + 6a4z2 - 5a4z4 + a4z6 + 2a5z - 4a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                   1 j = 5                     j = 3               1 1   j = 1           3 1       j = -1         1 4 1       j = -3       1 1 2         j = -5       1             j = -7   1 1 1             j = -9                     j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[9, NonAlternating, 27]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, NonAlternating, 27]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[9, 18, 10, 15], > X[8, 4, 9, 3], X[5, 17, 6, 16], X[17, 5, 18, 14], X[15, 10, 16, 11], > X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -5, -4, 8, 9, -2, 3, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 -4 -2 1 2 3 1 - q + q + q + - + q - q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 -10 3 4 5 4 2 4 6 8 4 - q - q - q + q + -- + -- + -- + -- + 3 q + 2 q + 2 q + q + 8 6 4 2 q q q q 10 > q
In[8]:=	HOMFLYPT[Link[9, NonAlternating, 27]][a, z]
Out[8]=	2 2 2 2 4 2 1 a 2 z 2 2 4 2 4 -6 + -- + 6 a - 2 a - -- + ----- + -- - 5 z + -- + 5 a z - a z - z + 2 2 2 2 2 2 a z a z z a 2 4 > a z
In[9]:=	Kauffman[Link[9, NonAlternating, 27]][a, z]
Out[9]=	2 4 2 4 2 1 a 2 2 a 2 z 3 -11 - -- - 12 a - 4 a + -- + ----- + -- - --- - --- + --- + 6 a z + 6 a z + 2 2 2 2 2 a z z a a z a z z 2 3 5 2 6 z 2 2 4 2 4 z 3 3 3 > 2 a z + 22 z + ---- + 22 a z + 6 a z + ---- + 2 a z - 6 a z - 2 a a 4 5 5 3 4 5 z 2 4 4 4 5 z 5 3 5 > 4 a z - 13 z - ---- - 13 a z - 5 a z - ---- - 5 a z + a z + 2 a a 6 7 5 5 6 z 2 6 4 6 z 7 > a z + 2 z + -- + 2 a z + a z + -- + a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 1 1 1 1 1 1 1 -- + - + 3 q + ------ + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q 11 5 7 4 7 3 7 2 5 2 3 2 3 q t q q t q t q t q t q t q t q t t 3 2 3 3 7 4 > - + q t + q t + q t + q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9n27

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