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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X18,16,11,15 X14,8,15,7 X10,12,5,11 X8,17,9,18 X16,9,17,10 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 5q-1 - 5 + 8q - 7q2 + 7q3 - 5q4 + 3q5 - q6

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 4q-4 + 3q-2 + 5 + 5q2 + 2q4 + 4q6 + 2q10 - q14 + q16 - q18

HOMFLY-PT Polynomial:

- a-4 - 2a-4z2 - a-4z4 + a-2z-2 + 4a-2 + 6a-2z2 + 4a-2z4 + a-2z6 - 2z-2 - 5 - 6z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

a-7z3 - a-6z2 + 3a-6z4 + a-5z - 4a-5z3 + 5a-5z5 - 2a-4 + 6a-4z2 - 9a-4z4 + 6a-4z6 + 3a-3z - 5a-3z3 - 3a-3z5 + 4a-3z7 + a-2z-2 - 8a-2 + 23a-2z2 - 27a-2z4 + 8a-2z6 + a-2z8 - 2a-1z-1 + 5a-1z + 3a-1z3 - 14a-1z5 + 6a-1z7 + 2z-2 - 9 + 22z2 - 19z4 + 3z6 + z8 - 2az-1 + 3az + 3az3 - 6az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 2   j = 9               3 1   j = 7             4 2     j = 5           4 4       j = 3         4 3         j = 1       3 6           j = -1     2 2             j = -3   1 4               j = -5   1                 j = -7 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, Alternating, 50]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 50]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 16, 11, 15], X[14, 8, 15, 7], > X[10, 12, 5, 11], X[8, 17, 9, 18], X[16, 9, 17, 10], X[2, 5, 3, 6], > X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -6, 7, -5}, {5, -2, 9, -4, 3, -7, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 5 2 3 4 5 6 -5 + q - -- + - + 8 q - 7 q + 7 q - 5 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 4 3 2 4 6 10 14 16 18 5 + q + q + q + -- + -- + 5 q + 2 q + 4 q + 2 q - q + q - q 4 2 q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 50]][a, z]
Out[8]=	2 2 2 -4 4 2 2 1 a 2 2 z 6 z 2 2 4 -5 - a + -- + 2 a - -- + ----- + -- - 6 z - ---- + ---- + a z - 2 z - 2 2 2 2 2 4 2 a z a z z a a 4 4 6 z 4 z z > -- + ---- + -- 4 2 2 a a a
In[9]:=	Kauffman[Link[9, Alternating, 50]][a, z]
Out[9]=	2 2 8 2 2 1 a 2 2 a z 3 z 5 z -9 - -- - -- - 4 a + -- + ----- + -- - --- - --- + -- + --- + --- + 3 a z + 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 3 2 z 6 z 23 z 2 2 z 4 z 5 z 3 z 3 > 22 z - -- + ---- + ----- + 6 a z + -- - ---- - ---- + ---- + 3 a z - 6 4 2 7 5 3 a a a a a a a 4 4 4 5 5 5 4 3 z 9 z 27 z 2 4 5 z 3 z 14 z 5 > 19 z + ---- - ---- - ----- - 4 a z + ---- - ---- - ----- - 6 a z + 6 4 2 5 3 a a a a a a 6 6 7 7 8 6 6 z 8 z 2 6 4 z 6 z 7 8 z > 3 z + ---- + ---- + a z + ---- + ---- + 2 a z + z + -- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 4 2 2 3 q 3 6 q + 4 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 3 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 4 q t + 4 q t + 4 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 L9a50

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