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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,16,8,17 X20,12,21,11 X22,18,5,17 X18,22,19,21 X10,20,11,19 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13

Gauss Code:

{{1, -10, 2, -11}, {10, -1, -3, 9, -8, -7, 4, -2, 11, 8, -9, 3, 5, -6, 7, -4, 6, -5}}

Jones Polynomial:

q-5/2 - 4q-3/2 + 6q-1/2 - 10q1/2 + 11q3/2 - 11q5/2 + 9q7/2 - 7q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-8 + 2q-6 + q-4 + 5 + 2q4 - q8 + 2q10 - 2q12 + 2q14 - 2q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 7a-6z4 - 4a-6z6 - 4a-5z3 + 12a-5z5 - 6a-5z7 - 2a-4z2 + 3a-4z4 + 4a-4z6 - 4a-4z8 - 6a-3z3 + 13a-3z5 - 6a-3z7 - a-3z9 - 6a-2z4 + 8a-2z6 - 5a-2z8 - a-1z-1 + 4a-1z3 - 4a-1z5 - a-1z9 + 1 + 2z2 - 3z4 - z8 - az-1 + 5az3 - 4az5 + a2z2 - a2z4

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                   1 j = 12                 3   j = 10               4 1   j = 8             5 3     j = 6           6 4       j = 4         5 5         j = 2       5 6           j = 0     3 7             j = -2   1 3               j = -4   3                 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[11, NonAlternating, 92]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 92]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 16, 8, 17], X[20, 12, 21, 11], > X[22, 18, 5, 17], X[18, 22, 19, 21], X[10, 20, 11, 19], X[9, 14, 10, 15], > X[15, 8, 16, 9], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, -3, 9, -8, -7, 4, -2, 11, 8, -9, 3, 5, -6, > 7, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 4 6 3/2 5/2 7/2 9/2 q - ---- + ------- - 10 Sqrt[q] + 11 q - 11 q + 9 q - 7 q + 3/2 Sqrt[q] q 11/2 13/2 > 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 -4 4 8 10 12 14 18 20 5 - q + -- + q + 2 q - q + 2 q - 2 q + 2 q - 2 q + q 6 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 92]][a, z]
Out[8]=	3 3 3 5 5 1 a z z z 3 z z -(---) + - - -- + -- + -- - a z + -- + -- a z z 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 92]][a, z]
Out[9]=	2 2 3 3 3 3 1 a 2 z 2 z 2 2 z 4 z 6 z 4 z 3 1 - --- - - + 2 z - -- - ---- + a z + -- - ---- - ---- + ---- + 5 a z - a z z 6 4 7 5 3 a a a a a a 4 4 4 5 5 5 5 4 7 z 3 z 6 z 2 4 z 12 z 13 z 4 z 5 > 3 z + ---- + ---- - ---- - a z - -- + ----- + ----- - ---- - 4 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 8 8 9 9 4 z 4 z 8 z 6 z 6 z 8 4 z 5 z z z > ---- + ---- + ---- - ---- - ---- - z - ---- - ---- - -- - -- 6 4 2 5 3 4 2 3 a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 3 3 2 4 4 2 7 + 5 q + ----- + ----- + ----- + - + ---- + 6 q t + 5 q t + 5 q t + 6 3 4 2 2 2 t 2 q t q t q t q t 6 2 6 3 8 3 8 4 10 4 10 5 12 5 > 6 q t + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + 14 6 > q t

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