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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-15/2 + 2q-13/2 - 4q-11/2 + 6q-9/2 - 6q-7/2 + 5q-5/2 - 6q-3/2 + 3q-1/2 - 2q1/2 + q3/2

A2 (sl(3)) Invariant:

q-22 + 2q-18 - q-14 - q-10 + 4q-8 + 2q-6 + 3q-4 + q-2 - 1 - q4

HOMFLY-PT Polynomial:

4az + 4az3 + az5 - a3z-1 - 11a3z - 13a3z3 - 6a3z5 - a3z7 + a5z-1 + 5a5z + 4a5z3 + a5z5

Kauffman Polynomial:

- 4z2 + 4z4 - z6 + 5az - 10az3 + 8az5 - 2az7 - 7a2z2 + 8a2z4 - a2z8 - a3z-1 + 11a3z - 17a3z3 + 16a3z5 - 5a3z7 + a4 - 10a4z2 + 13a4z4 - 3a4z6 - a4z8 - a5z-1 + 4a5z - 4a5z3 + 5a5z5 - 3a5z7 - 6a6z2 + 7a6z4 - 4a6z6 - a7z + 2a7z3 - 3a7z5 + a8z2 - 2a8z4 + a9z - a9z3

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                   1 j = 2                 1   j = 0               2 1   j = -2             4 1     j = -4           2 3       j = -6         4 3         j = -8       2 2           j = -10     2 4             j = -12   1 3               j = -14   1                 j = -16 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[9, Alternating, 41]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 41]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[18, 5, 9, 6], > X[14, 8, 15, 7], X[16, 14, 17, 13], X[8, 16, 1, 15], X[6, 9, 7, 10], > X[4, 17, 5, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -9, 4, -8, 5, -7}, {8, -1, 2, -3, 6, -5, 7, -6, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 4 6 6 5 6 3 -q + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 2 Sqrt[q] + 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -14 -10 4 2 3 -2 4 -1 + q + --- - q - q + -- + -- + -- + q - q 18 8 6 4 q q q q
In[8]:=	HOMFLYPT[Link[9, Alternating, 41]][a, z]
Out[8]=	3 5 a a 3 5 3 3 3 5 3 5 -(--) + -- + 4 a z - 11 a z + 5 a z + 4 a z - 13 a z + 4 a z + a z - z z 3 5 5 5 3 7 > 6 a z + a z - a z
In[9]:=	Kauffman[Link[9, Alternating, 41]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 2 2 2 a - -- - -- + 5 a z + 11 a z + 4 a z - a z + a z - 4 z - 7 a z - z z 4 2 6 2 8 2 3 3 3 5 3 7 3 > 10 a z - 6 a z + a z - 10 a z - 17 a z - 4 a z + 2 a z - 9 3 4 2 4 4 4 6 4 8 4 5 3 5 > a z + 4 z + 8 a z + 13 a z + 7 a z - 2 a z + 8 a z + 16 a z + 5 5 7 5 6 4 6 6 6 7 3 7 5 7 > 5 a z - 3 a z - z - 3 a z - 4 a z - 2 a z - 5 a z - 3 a z - 2 8 4 8 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 1 1 3 2 4 2 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 4 2 16 6 14 5 12 5 12 4 10 4 10 3 8 3 8 2 q q q t q t q t q t q t q t q t q t 4 3 2 t 2 2 2 4 3 > ----- + ---- + ---- + 2 t + -- + t + q t + q t 6 2 6 4 2 q t q t q t q

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