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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X20,14,21,13 X22,17,11,18 X18,21,19,22 X6,16,7,15 X16,8,17,7 X14,20,15,19 X10,6,1,5 X4,10,5,9 X2,11,3,12

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 6q-1/2 + 10q1/2 - 14q3/2 + 16q5/2 - 17q7/2 + 15q9/2 - 12q11/2 + 7q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 4q-2 - 1 + q2 - 4q6 + 3q8 - 2q10 + 4q12 + 2q14 + 3q18 - 3q20 - q26

HOMFLY-PT Polynomial:

a-7z-1 + 2a-7z + a-7z3 - 4a-5z-1 - 7a-5z - 6a-5z3 - 2a-5z5 + 6a-3z-1 + 11a-3z + 8a-3z3 + 4a-3z5 + a-3z7 - 5a-1z-1 - 9a-1z - 7a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

a-10z2 - a-10z4 + 2a-9z3 - 3a-9z5 + a-8 - 4a-8z2 + 6a-8z4 - 6a-8z6 - a-7z-1 + 6a-7z - 15a-7z3 + 15a-7z5 - 9a-7z7 + 3a-6 - 9a-6z2 + 3a-6z4 + 9a-6z6 - 8a-6z8 - 4a-5z-1 + 20a-5z - 41a-5z3 + 37a-5z5 - 7a-5z7 - 4a-5z9 + 3a-4 - 6a-4z2 - 10a-4z4 + 28a-4z6 - 11a-4z8 - a-4z10 - 6a-3z-1 + 29a-3z - 48a-3z3 + 31a-3z5 + 4a-3z7 - 6a-3z9 + a-2 - a-2z2 - 13a-2z4 + 20a-2z6 - 5a-2z8 - a-2z10 - 5a-1z-1 + 22a-1z - 33a-1z3 + 17a-1z5 + a-1z7 - 2a-1z9 + 1 + z2 - 7z4 + 7z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - az7

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 r = 7 j = 18                       1 j = 16                     2   j = 14                   5 1   j = 12                 7 2     j = 10               8 5       j = 8             9 7         j = 6           7 8           j = 4         7 9             j = 2       5 9               j = 0     1 5                 j = -2   1 5                   j = -4   1                     j = -6 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, Alternating, 346]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 346]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[20, 14, 21, 13], X[22, 17, 11, 18], > X[18, 21, 19, 22], X[6, 16, 7, 15], X[16, 8, 17, 7], X[14, 20, 15, 19], > X[10, 6, 1, 5], X[4, 10, 5, 9], X[2, 11, 3, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 9, -6, 7, -2, 10, -9}, > {11, -1, 3, -8, 6, -7, 4, -5, 8, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 6 3/2 5/2 7/2 -q + ---- - ------- + 10 Sqrt[q] - 14 q + 16 q - 17 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 15 q - 12 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 4 2 6 8 10 12 14 18 -1 + q + q + q + -- + q - 4 q + 3 q - 2 q + 4 q + 2 q + 3 q - 2 q 20 26 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 346]][a, z]
Out[8]=	3 3 1 4 6 5 2 a 2 z 7 z 11 z 9 z z 6 z ---- - ---- + ---- - --- + --- + --- - --- + ---- - --- + 3 a z + -- - ---- + 7 5 3 a z z 7 5 3 a 7 5 a z a z a z a a a a a 3 3 5 5 5 7 8 z 7 z 3 2 z 4 z 2 z z > ---- - ---- + a z - ---- + ---- - ---- + -- 3 a 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 346]][a, z]
Out[9]=	-8 3 3 -2 1 4 6 5 2 a 6 z 20 z 29 z 1 + a + -- + -- + a - ---- - ---- - ---- - --- - --- + --- + ---- + ---- + 6 4 7 5 3 a z z 7 5 3 a a a z a z a z a a a 2 2 2 2 2 3 3 3 22 z 2 z 4 z 9 z 6 z z 2 z 15 z 41 z > ---- + 7 a z + z + --- - ---- - ---- - ---- - -- + ---- - ----- - ----- - a 10 8 6 4 2 9 7 5 a a a a a a a a 3 3 4 4 4 4 4 5 48 z 33 z 3 4 z 6 z 3 z 10 z 13 z 3 z > ----- - ----- - 9 a z - 7 z - --- + ---- + ---- - ----- - ----- - ---- + 3 a 10 8 6 4 2 9 a a a a a a a 5 5 5 5 6 6 6 15 z 37 z 31 z 17 z 5 6 6 z 9 z 28 z > ----- + ----- + ----- + ----- + 5 a z + 7 z - ---- + ---- + ----- + 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 8 8 20 z 9 z 7 z 4 z z 7 8 8 z 11 z 5 z > ----- - ---- - ---- + ---- + -- - a z - 2 z - ---- - ----- - ---- - 2 7 5 3 a 6 4 2 a a a a a a a 9 9 9 10 10 4 z 6 z 2 z z z > ---- - ---- - ---- - --- - --- 5 3 a 4 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 5 5 5 q 4 9 q + 7 q + ----- + ----- + ----- + t + ----- + - + ---- + 9 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 7 q t + 8 q t + 9 q t + 7 q t + 8 q t + 5 q t + 7 q t + 12 5 14 5 14 6 16 6 18 7 > 2 q t + 5 q t + q t + 2 q t + q t

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