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DrawMorseLink

PD Presentation:

X12,1,13,2 X3849 X5,14,6,15 X7,18,8,19 X9,21,10,20 X10,11,1,12 X13,6,14,7 X17,4,18,5 X15,11,16,22 X19,3,20,2 X21,17,22,16

Gauss Code:

{{1, 10, -2, 8, -3, 7, -4, 2, -5, -6}, {6, -1, -7, 3, -9, 11, -8, 4, -10, 5, -11, 9}}

Jones Polynomial:

q-15/2 - 4q-13/2 + 7q-11/2 - 10q-9/2 + 11q-7/2 - 12q-5/2 + 10q-3/2 - 8q-1/2 + 4q1/2 - q3/2

A2 (sl(3)) Invariant:

- q-24 + 3q-20 - q-18 + 2q-16 + q-14 - 2q-12 + 3q-10 + 4q-6 + q-4 - q-2 + 2 - 3q2 + q6

HOMFLY-PT Polynomial:

- a-1z + 3az + 3az3 - a3z-1 - 7a3z - 6a3z3 - 2a3z5 + a5z-1 + 4a5z + 3a5z3 - a7z

Kauffman Polynomial:

a-1z - a-1z3 + 2z2 - 4z4 + 3az - 7az3 + 2az5 - 2az7 + 4a2z2 - 14a2z4 + 10a2z6 - 4a2z8 - a3z-1 + 8a3z - 19a3z3 + 15a3z5 - 2a3z7 - 2a3z9 + a4 + 2a4z2 - 15a4z4 + 23a4z6 - 9a4z8 - a5z-1 + 7a5z - 20a5z3 + 24a5z5 - 4a5z7 - 2a5z9 - a6z2 - 3a6z4 + 12a6z6 - 5a6z8 + a7z - 7a7z3 + 11a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 4                   1 j = 2                 3   j = 0               5 1   j = -2             6 4     j = -4           6 4       j = -6         5 6         j = -8       5 6           j = -10     3 6             j = -12   1 4               j = -14   3                 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[11, NonAlternating, 246]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 246]]
Out[4]=	PD[X[12, 1, 13, 2], X[3, 8, 4, 9], X[5, 14, 6, 15], X[7, 18, 8, 19], > X[9, 21, 10, 20], X[10, 11, 1, 12], X[13, 6, 14, 7], X[17, 4, 18, 5], > X[15, 11, 16, 22], X[19, 3, 20, 2], X[21, 17, 22, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 10, -2, 8, -3, 7, -4, 2, -5, -6}, > {6, -1, -7, 3, -9, 11, -8, 4, -10, 5, -11, 9}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 7 10 11 12 10 8 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 4 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]=	-24 3 -18 2 -14 2 3 4 -4 -2 2 6 2 - q + --- - q + --- + q - --- + --- + -- + q - q - 3 q + q 20 16 12 10 6 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 246]][a, z]
Out[8]=	3 5 a a z 3 5 7 3 3 3 5 3 -(--) + -- - - + 3 a z - 7 a z + 4 a z - a z + 3 a z - 6 a z + 3 a z - z z a 3 5 > 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 246]][a, z]
Out[9]=	3 5 4 a a z 3 5 7 2 2 2 4 2 a - -- - -- + - + 3 a z + 8 a z + 7 a z + a z + 2 z + 4 a z + 2 a z - z z a 3 6 2 8 2 z 3 3 3 5 3 7 3 4 > a z - a z - -- - 7 a z - 19 a z - 20 a z - 7 a z - 4 z - a 2 4 4 4 6 4 8 4 5 3 5 5 5 > 14 a z - 15 a z - 3 a z + 2 a z + 2 a z + 15 a z + 24 a z + 7 5 2 6 4 6 6 6 8 6 7 3 7 > 11 a z + 10 a z + 23 a z + 12 a z - a z - 2 a z - 2 a z - 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 4 a z - 4 a z - 4 a z - 9 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 3 1 4 3 6 5 6 5 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 6 6 4 6 2 4 2 > ----- + ----- + ----- + ---- + ---- + t + 3 q t + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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