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DrawMorseLink

PD Presentation:

X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X20,17,11,18 X18,13,19,14 X14,19,15,20 X2,11,3,12 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-24 - 2q-22 - q-20 - 2q-18 + 2q-16 + 4q-14 + 4q-12 + 6q-10 + 2q-8 + 2q-6 - q-4 - 2q-2 - 2q2

HOMFLY-PT Polynomial:

2az-1 + 4az + 2az3 - 7a3z-1 - 14a3z - 9a3z3 - 2a3z5 + 7a5z-1 + 9a5z + 3a5z3 - 2a7z-1 - a7z

Kauffman Polynomial:

2 - 3z2 - 2az-1 + 3az - 3az3 - az5 + 8a2 - 17a2z2 + 11a2z4 - 4a2z6 - 7a3z-1 + 16a3z - 18a3z3 + 12a3z5 - 4a3z7 + 13a4 - 29a4z2 + 24a4z4 - 5a4z6 - a4z8 - 7a5z-1 + 17a5z - 22a5z3 + 20a5z5 - 6a5z7 + 8a6 - 20a6z2 + 17a6z4 - 2a6z6 - a6z8 - 2a7z-1 + 4a7z - 7a7z3 + 7a7z5 - 2a7z7 + 2a8 - 5a8z2 + 4a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 j = 2                 2 j = 0               2   j = -2             4 3   j = -4           3 1     j = -6         2 4       j = -8       4 3         j = -10     1 3           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[10, NonAlternating, 61]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 61]]
Out[4]=	PD[X[12, 1, 13, 2], X[16, 7, 17, 8], X[5, 1, 6, 10], X[3, 7, 4, 6], > X[9, 5, 10, 4], X[20, 17, 11, 18], X[18, 13, 19, 14], X[14, 19, 15, 20], > X[2, 11, 3, 12], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, > {9, -1, 7, -8, 10, -2, 6, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 4 6 5 7 5 4 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 2 Sqrt[q] 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -20 2 2 4 4 6 2 2 -4 2 2 -q - --- - q - --- + --- + --- + --- + --- + -- + -- - q - -- - 2 q 22 18 16 14 12 10 8 6 2 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 61]][a, z]
Out[8]=	3 5 7 2 a 7 a 7 a 2 a 3 5 7 3 3 3 --- - ---- + ---- - ---- + 4 a z - 14 a z + 9 a z - a z + 2 a z - 9 a z + z z z z 5 3 3 5 > 3 a z - 2 a z
In[9]:=	Kauffman[Link[10, NonAlternating, 61]][a, z]
Out[9]=	3 5 7 2 4 6 8 2 a 7 a 7 a 2 a 3 2 + 8 a + 13 a + 8 a + 2 a - --- - ---- - ---- - ---- + 3 a z + 16 a z + z z z z 5 7 2 2 2 4 2 6 2 8 2 > 17 a z + 4 a z - 3 z - 17 a z - 29 a z - 20 a z - 5 a z - 3 3 3 5 3 7 3 2 4 4 4 6 4 > 3 a z - 18 a z - 22 a z - 7 a z + 11 a z + 24 a z + 17 a z + 8 4 5 3 5 5 5 7 5 2 6 4 6 > 4 a z - a z + 12 a z + 20 a z + 7 a z - 4 a z - 5 a z - 6 6 8 6 3 7 5 7 7 7 4 8 6 8 > 2 a z - a z - 4 a z - 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 3 1 3 4 3 2 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 2 4 3 1 4 2 > ----- + ----- + ----- + ---- + ---- + 2 q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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