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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,17,8,16 X20,9,11,10 X18,12,19,11 X15,9,16,8 X10,19,5,20 X17,14,18,15 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-3 + 4q-2 - 4q-1 + 7 - 6q + 6q2 - 4q3 + 3q4 - q5

A2 (sl(3)) Invariant:

- q-10 + 2q-8 + 3q-6 + 2q-4 + 6q-2 + 4 + 5q2 + 3q4 + q6 + 2q8 - q10 + q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z-2 + a-2 + a-2z2 + a-2z4 - 2z-2 - 2 + z4 + a2z-2 + a2 - a2z2

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 4a-4z2 - 8a-4z4 + 3a-4z6 + 3a-3z3 - 7a-3z5 + 3a-3z7 + a-2z-2 - 2a-2 + 5a-2z2 - 8a-2z4 + 2a-2z6 + a-2z8 - 2a-1z-1 + 2a-1z + 5a-1z3 - 8a-1z5 + 4a-1z7 + 2z-2 - 3 - 2z2 + 4z4 - z6 + z8 - 2az-1 + 2az + az3 + az7 + a2z-2 - 2a2 - 3a2z2 + 4a2z4 + a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                 1 j = 9               2   j = 7             2 1   j = 5           4 2     j = 3         3 3       j = 1       4 3         j = -1     2 5           j = -3   2 2             j = -5   3               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[10, NonAlternating, 73]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 73]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 17, 8, 16], X[20, 9, 11, 10], > X[18, 12, 19, 11], X[15, 9, 16, 8], X[10, 19, 5, 20], X[17, 14, 18, 15], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -3, 6, 4, -7}, > {5, -2, 10, 8, -6, 3, -8, -5, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-3 4 4 2 3 4 5 7 - q + -- - - - 6 q + 6 q - 4 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 3 2 6 2 4 6 8 10 12 14 16 4 - q + -- + -- + -- + -- + 5 q + 3 q + q + 2 q - q + q + q - q 8 6 4 2 q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 73]][a, z]
Out[8]=	2 2 2 4 -2 2 2 1 a z z 2 2 4 z -2 + a + a - -- + ----- + -- - -- + -- - a z + z + -- 2 2 2 2 4 2 2 z a z z a a a
In[9]:=	Kauffman[Link[10, NonAlternating, 73]][a, z]
Out[9]=	2 2 2 2 2 1 a 2 2 a 2 z 2 4 z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z - 2 z + ---- + 2 2 2 2 2 a z z a 4 a z a z z a 2 3 3 3 4 4 5 z 2 2 2 z 3 z 5 z 3 3 3 4 8 z 8 z > ---- - 3 a z - ---- + ---- + ---- + a z + a z + 4 z - ---- - ---- + 2 5 3 a 4 2 a a a a a 5 5 5 6 6 7 7 8 2 4 z 7 z 8 z 6 3 z 2 z 3 z 4 z 7 8 z > 4 a z + -- - ---- - ---- - z + ---- + ---- + ---- + ---- + a z + z + -- 5 3 a 4 2 3 a 2 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 3 2 2 2 3 3 2 - + 4 q + ----- + ----- + ----- + ---- + --- + 3 q t + 3 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 11 5 > 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

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