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DrawMorseLink

PD Presentation:

X6172 X14,3,15,4 X18,8,19,7 X20,10,5,9 X8,20,9,19 X16,12,17,11 X12,16,13,15 X10,18,11,17 X2536 X4,13,1,14

Gauss Code:

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

Jones Polynomial:

- q-7/2 + q-5/2 - 4q-3/2 + 5q-1/2 - 8q1/2 + 9q3/2 - 8q5/2 + 7q7/2 - 5q9/2 + 3q11/2 - q13/2

A2 (sl(3)) Invariant:

q-12 + 2q-10 + 2q-8 + 4q-6 + 2q-4 + 2 - 2q2 - q6 - q8 + q10 - 2q12 + q14 - q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

2a-7z3 - a-7z5 - 2a-6z2 + 7a-6z4 - 3a-6z6 + a-5z - 6a-5z3 + 10a-5z5 - 4a-5z7 + 3a-4z2 - 7a-4z4 + 7a-4z6 - 3a-4z8 - 4a-3z3 + 4a-3z5 - a-3z7 - a-3z9 - a-2 + 7a-2z2 - 16a-2z4 + 11a-2z6 - 4a-2z8 + a-1z-1 - 3a-1z + 7a-1z3 - 7a-1z5 + 2a-1z7 - a-1z9 - 3 + 5z2 - z4 - z8 + 3az-1 - 7az + 7az3 - az5 - az7 - 3a2 + 3a2z2 + a2z4 - a2z6 + 2a3z-1 - 5a3z + 4a3z3 - 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                   2   j = 10                 3 1   j = 8               4 2     j = 6             4 3       j = 4           5 4         j = 2         3 4           j = 0       3 6             j = -2     1 2               j = -4     3                 j = -6 1 1                   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, 47]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 47]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[18, 8, 19, 7], X[20, 10, 5, 9], > X[8, 20, 9, 19], X[16, 12, 17, 11], X[12, 16, 13, 15], X[10, 18, 11, 17], > X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -5, 4, -8, 6, -7, 10, -2, 7, -6, 8, -3, > 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) -(5/2) 4 5 3/2 5/2 7/2 -q + q - ---- + ------- - 8 Sqrt[q] + 9 q - 8 q + 7 q - 3/2 Sqrt[q] q 9/2 11/2 13/2 > 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 2 4 2 2 6 8 10 12 14 18 20 2 + q + --- + -- + -- + -- - 2 q - q - q + q - 2 q + q - q + q 10 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 47]][a, z]
Out[8]=	3 3 3 3 1 3 a 2 a z z 2 z 3 z 2 z 2 z 3 --- - --- + ---- - -- + -- + --- - 5 a z + a z - -- + ---- + ---- - 2 a z + a z z z 5 3 a 5 3 a a a a a 5 5 z z > -- + -- 3 a a
In[9]:=	Kauffman[Link[10, Alternating, 47]][a, z]
Out[9]=	3 2 -2 2 1 3 a 2 a z 3 z 3 2 2 z -3 - a - 3 a + --- + --- + ---- + -- - --- - 7 a z - 5 a z + 5 z - ---- + a z z z 5 a 6 a a 2 2 3 3 3 3 3 z 7 z 2 2 2 z 6 z 4 z 7 z 3 3 3 4 > ---- + ---- + 3 a z + ---- - ---- - ---- + ---- + 7 a z + 4 a z - z + 4 2 7 5 3 a a a a a a 4 4 4 5 5 5 5 7 z 7 z 16 z 2 4 z 10 z 4 z 7 z 5 3 5 > ---- - ---- - ----- + a z - -- + ----- + ---- - ---- - a z - a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 8 8 3 z 7 z 11 z 2 6 4 z z 2 z 7 8 3 z 4 z > ---- + ---- + ----- - a z - ---- - -- + ---- - a z - z - ---- - ---- - 6 4 2 5 3 a 4 2 a a a a a a a 9 9 z z > -- - -- 3 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 1 3 1 3 2 2 4 6 + 3 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 4 q t + 5 q t + 8 4 6 4 6 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 > 4 q t + 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 12 5 14 6 > 2 q t + q t

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