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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X18,10,7,9 X2738 X16,13,17,14 X6,12,1,11 X4,16,5,15 X14,6,15,5 X12,17,13,18

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 5q-3/2 + 7q-1/2 - 9q1/2 + 8q3/2 - 8q5/2 + 5q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-12 - 2q-8 + q-6 - q-4 + 3 + 3q4 + 2q8 + 2q10 - q12 + 2q14 - q18

HOMFLY-PT Polynomial:

a-5z - a-3z-1 - 2a-3z - 2a-3z3 + a-1z-1 + 3a-1z + 2a-1z3 + a-1z5 - 2az - 2az3 + a3z

Kauffman Polynomial:

a-6z2 - a-6z4 - 2a-5z + 4a-5z3 - 3a-5z5 - a-4z2 + 4a-4z4 - 4a-4z6 + a-3z-1 - 4a-3z + 6a-3z3 - a-3z5 - 3a-3z7 - a-2 - 5a-2z2 + 13a-2z4 - 7a-2z6 - a-2z8 + a-1z-1 - 2a-1z + a-1z3 + 7a-1z5 - 6a-1z7 - 7z2 + 15z4 - 6z6 - z8 - az + az3 + 4az5 - 3az7 - 4a2z2 + 7a2z4 - 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 j = 12                   1 j = 10                 2   j = 8               3 1   j = 6             5 2     j = 4           4 4       j = 2         5 4         j = 0       3 5           j = -2     2 4             j = -4   1 3               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[9, Alternating, 27]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 27]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[18, 10, 7, 9], X[2, 7, 3, 8], > X[16, 13, 17, 14], X[6, 12, 1, 11], X[4, 16, 5, 15], X[14, 6, 15, 5], > X[12, 17, 13, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 8, -6}, {4, -1, 3, -2, 6, -9, 5, -8, 7, -5, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 5 7 3/2 5/2 7/2 -q + ---- - ---- + ------- - 9 Sqrt[q] + 8 q - 8 q + 5 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 -6 -4 4 8 10 12 14 18 3 + q - -- + q - q + 3 q + 2 q + 2 q - q + 2 q - q 8 q
In[8]:=	HOMFLYPT[Link[9, Alternating, 27]][a, z]
Out[8]=	3 3 5 1 1 z 2 z 3 z 3 2 z 2 z 3 z -(----) + --- + -- - --- + --- - 2 a z + a z - ---- + ---- - 2 a z + -- 3 a z 5 3 a 3 a a a z a a a
In[9]:=	Kauffman[Link[9, Alternating, 27]][a, z]
Out[9]=	2 2 2 -2 1 1 2 z 4 z 2 z 3 2 z z 5 z -a + ---- + --- - --- - --- - --- - a z - a z - 7 z + -- - -- - ---- - 3 a z 5 3 a 6 4 2 a z a a a a a 3 3 3 4 4 4 2 2 4 z 6 z z 3 3 3 4 z 4 z 13 z > 4 a z + ---- + ---- + -- + a z + 2 a z + 15 z - -- + ---- + ----- + 5 3 a 6 4 2 a a a a a 5 5 5 6 6 2 4 3 z z 7 z 5 3 5 6 4 z 7 z > 7 a z - ---- - -- + ---- + 4 a z - a z - 6 z - ---- - ---- - 5 3 a 4 2 a a a a 7 7 8 2 6 3 z 6 z 7 8 z > 3 a z - ---- - ---- - 3 a z - z - -- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 3 2 3 4 2 4 5 + 5 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 4 q t + 4 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 > 4 q t + 5 q t + 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 L9a27

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