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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X18,14,19,13 X20,8,11,7 X6,20,7,19 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 6q-1/2 + 8q1/2 - 12q3/2 + 11q5/2 - 11q7/2 + 10q9/2 - 6q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + 2q-2 - 1 + 3q2 + 3q4 + 2q6 + 4q8 - 3q10 - 3q14 - q16 + 2q18 - q20 + q22

HOMFLY-PT Polynomial:

- 3a-5z - 3a-5z3 - a-5z5 + 8a-3z + 10a-3z3 + 5a-3z5 + a-3z7 - a-1z-1 - 7a-1z - 7a-1z3 - 2a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

- a-9z3 - 3a-8z4 + 3a-7z3 - 6a-7z5 - 8a-6z2 + 16a-6z4 - 10a-6z6 + 6a-5z - 16a-5z3 + 24a-5z5 - 11a-5z7 - 11a-4z2 + 10a-4z4 + 10a-4z6 - 7a-4z8 + 16a-3z - 44a-3z3 + 44a-3z5 - 9a-3z7 - 2a-3z9 - 23a-2z4 + 32a-2z6 - 10a-2z8 - a-1z-1 + 14a-1z - 30a-1z3 + 18a-1z5 + a-1z7 - 2a-1z9 + 1 + 3z2 - 14z4 + 12z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - az7

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 = 16                     1 j = 14                   2   j = 12                 4 1   j = 10               6 2     j = 8             5 4       j = 6           6 6         j = 4         6 5           j = 2       4 8             j = 0     2 4               j = -2   1 4                 j = -4   2                   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[10, Alternating, 113]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 113]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[18, 14, 19, 13], X[20, 8, 11, 7], > X[6, 20, 7, 19], X[4, 15, 5, 16], X[14, 5, 15, 6], X[16, 10, 17, 9], > X[2, 11, 3, 12], X[10, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -6, 7, -5, 4, -2, 8, -10}, > {9, -1, 3, -7, 6, -8, 10, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 6 3/2 5/2 7/2 9/2 -q + ---- - ------- + 8 Sqrt[q] - 12 q + 11 q - 11 q + 10 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 2 2 4 6 8 10 14 16 18 20 -1 + q + -- + 3 q + 3 q + 2 q + 4 q - 3 q - 3 q - q + 2 q - q + 2 q 22 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 113]][a, z]
Out[8]=	3 3 3 5 5 1 a 3 z 8 z 7 z 3 z 10 z 7 z 3 z 5 z -(---) + - - --- + --- - --- + 2 a z - ---- + ----- - ---- + a z - -- + ---- - a z z 5 3 a 5 3 a 5 3 a a a a a a 5 7 2 z z > ---- + -- a 3 a
In[9]:=	Kauffman[Link[10, Alternating, 113]][a, z]
Out[9]=	2 2 3 3 1 a 6 z 16 z 14 z 2 8 z 11 z z 3 z 1 - --- - - + --- + ---- + ---- + 4 a z + 3 z - ---- - ----- - -- + ---- - a z z 5 3 a 6 4 9 7 a a a a a a 3 3 3 4 4 4 4 16 z 44 z 30 z 3 4 3 z 16 z 10 z 23 z > ----- - ----- - ----- - 6 a z - 14 z - ---- + ----- + ----- - ----- - 5 3 a 8 6 4 2 a a a a a a 5 5 5 5 6 6 6 6 z 24 z 44 z 18 z 5 6 10 z 10 z 32 z > ---- + ----- + ----- + ----- + 4 a z + 12 z - ----- + ----- + ----- - 7 5 3 a 6 4 2 a a a a a a 7 7 7 8 8 9 9 11 z 9 z z 7 8 7 z 10 z 2 z 2 z > ----- - ---- + -- - a z - 3 z - ---- - ----- - ---- - ---- 5 3 a 4 2 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 4 4 4 q 4 6 8 q + 6 q + ----- + ----- + ----- + -- + ----- + - + ---- + 5 q t + 6 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 12 5 > 6 q t + 5 q t + 4 q t + 6 q t + 2 q t + 4 q t + q t + 14 5 16 6 > 2 q t + q t

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