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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 8q-7/2 - 11q-5/2 + 11q-3/2 - 11q-1/2 + 8q1/2 - 5q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 + 2q-14 - q-12 + 3q-10 + 2q-8 + q-6 + 3q-4 - 2q-2 + 2 - 2q2 - q4 + q6 - q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z - 3a-1z3 - a-1z5 + 5az + 9az3 + 5az5 + az7 - a3z-1 - 7a3z - 7a3z3 - 2a3z5 + a5z-1 + 2a5z + a5z3

Kauffman Polynomial:

2a-3z3 - a-3z5 - 3a-2z2 + 7a-2z4 - 3a-2z6 + 3a-1z - 7a-1z3 + 9a-1z5 - 4a-1z7 - 3z2 + 2z4 + 3z6 - 3z8 + 7az - 22az3 + 19az5 - 6az7 - az9 - 8a2z4 + 11a2z6 - 6a2z8 - a3z-1 + 8a3z - 18a3z3 + 17a3z5 - 6a3z7 - a3z9 + a4 - 2a4z2 + 3a4z4 + 2a4z6 - 3a4z8 - a5z-1 + 3a5z - 3a5z3 + 7a5z5 - 4a5z7 - 2a6z2 + 6a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:

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

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