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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X4,20,5,19 X14,6,15,5 X16,7,17,8 X18,16,19,15 X6,17,7,18 X8,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 6q-3/2 + 9q-1/2 - 13q1/2 + 13q3/2 - 13q5/2 + 10q7/2 - 7q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

q-12 - 2q-8 + 2q-6 + 5 + 3q4 + 2q10 - 3q12 + 2q14 - 2q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 7a-6z4 - 4a-6z6 + a-5z - 5a-5z3 + 11a-5z5 - 6a-5z7 - 7a-4z2 + 11a-4z4 - 4a-4z8 + 6a-3z - 20a-3z3 + 26a-3z5 - 11a-3z7 - a-3z9 - 7a-2z2 + 6a-2z4 + 6a-2z6 - 7a-2z8 - a-1z-1 + 8a-1z - 19a-1z3 + 21a-1z5 - 9a-1z7 - a-1z9 + 1 - 5z2 + 8z4 - z6 - 3z8 - az-1 + 2az - 3az3 + 6az5 - 4az7 - 3a2z2 + 6a2z4 - 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 r = 6 j = 14                     1 j = 12                   3   j = 10                 4 1   j = 8               6 3     j = 6             7 4       j = 4           6 6         j = 2         7 7           j = 0       4 8             j = -2     2 5               j = -4   1 4                 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[10, Alternating, 91]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 91]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[20, 12, 9, 11], X[2, 9, 3, 10], > X[4, 20, 5, 19], X[14, 6, 15, 5], X[16, 7, 17, 8], X[18, 16, 19, 15], > X[6, 17, 7, 18], X[8, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -9, 7, -10}, > {4, -1, 3, -2, 10, -6, 8, -7, 9, -8, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 6 9 3/2 5/2 7/2 -q + ---- - ---- + ------- - 13 Sqrt[q] + 13 q - 13 q + 10 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 7 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 2 4 10 12 14 18 20 5 + q - -- + -- + 3 q + 2 q - 3 q + 2 q - 2 q + q 8 6 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 91]][a, z]
Out[8]=	3 3 5 5 1 a z z 3 z z 3 z z -(---) + - + -- - - - a z + a z - -- + -- - 2 a z + -- + -- a z z 3 a 5 3 3 a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 91]][a, z]
Out[9]=	2 2 2 1 a z 6 z 8 z 3 2 2 z 7 z 7 z 1 - --- - - + -- + --- + --- + 2 a z - a z - 5 z - ---- - ---- - ---- - a z z 5 3 a 6 4 2 a a a a a 3 3 3 3 4 2 2 z 5 z 20 z 19 z 3 3 3 4 7 z > 3 a z + -- - ---- - ----- - ----- - 3 a z + 2 a z + 8 z + ---- + 7 5 3 a 6 a a a a 4 4 5 5 5 5 11 z 6 z 2 4 z 11 z 26 z 21 z 5 3 5 6 > ----- + ---- + 6 a z - -- + ----- + ----- + ----- + 6 a z - a z - z - 4 2 7 5 3 a a a a a a 6 6 7 7 7 8 8 4 z 6 z 2 6 6 z 11 z 9 z 7 8 4 z 7 z > ---- + ---- - 3 a z - ---- - ----- - ---- - 4 a z - 3 z - ---- - ---- - 6 2 5 3 a 4 2 a a a a a a 9 9 z z > -- - -- 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 4 5 2 4 8 + 7 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 7 q t + 6 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 10 5 > 6 q t + 7 q t + 4 q t + 6 q t + 3 q t + 4 q t + q t + 12 5 14 6 > 3 q t + q t

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