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DrawMorseLink

PD Presentation:

X12,1,13,2 X14,4,15,3 X22,7,11,8 X8,11,9,12 X20,14,21,13 X16,6,17,5 X18,10,19,9 X4,16,5,15 X6,18,7,17 X10,20,1,19 X2,21,3,22

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 6q-1/2 + 9q1/2 - 13q3/2 + 14q5/2 - 15q7/2 + 13q9/2 - 10q11/2 + 6q13/2 - 3q15/2 + q17/2

A2 (sl(3)) Invariant:

q-6 - q-4 + 2q-2 - 1 + q2 + 2q4 - q6 + 5q8 - 2q10 + 3q12 + 2q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial:

4a-5z + 8a-5z3 + 5a-5z5 + a-5z7 - a-3z-1 - 8a-3z - 20a-3z3 - 18a-3z5 - 7a-3z7 - a-3z9 + a-1z-1 + 5a-1z + 8a-1z3 + 5a-1z5 + a-1z7

Kauffman Polynomial:

a-10z2 - a-10z4 - a-9z + 3a-9z3 - 3a-9z5 - 2a-8z2 + 5a-8z4 - 5a-8z6 - a-7z - a-7z3 + 6a-7z5 - 6a-7z7 - 3a-6z4 + 8a-6z6 - 6a-6z8 - 4a-5z + 11a-5z3 - 11a-5z5 + 10a-5z7 - 5a-5z9 + 6a-4z2 - 16a-4z4 + 16a-4z6 - 2a-4z8 - 2a-4z10 + a-3z-1 - 9a-3z + 27a-3z3 - 40a-3z5 + 32a-3z7 - 9a-3z9 - a-2 + 7a-2z2 - 20a-2z4 + 15a-2z6 + a-2z8 - 2a-2z10 + a-1z-1 - 4a-1z + 8a-1z3 - 16a-1z5 + 15a-1z7 - 4a-1z9 + 4z2 - 13z4 + 12z6 - 3z8 + az - 4az3 + 4az5 - 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                   4 1   j = 12                 6 2     j = 10               7 4       j = 8             8 6         j = 6           7 8           j = 4         6 7             j = 2       4 8               j = 0     2 5                 j = -2   1 4                   j = -4   2                     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, 377]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 377]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 4, 15, 3], X[22, 7, 11, 8], X[8, 11, 9, 12], > X[20, 14, 21, 13], X[16, 6, 17, 5], X[18, 10, 19, 9], X[4, 16, 5, 15], > X[6, 18, 7, 17], X[10, 20, 1, 19], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -8, 6, -9, 3, -4, 7, -10}, > {4, -1, 5, -2, 8, -6, 9, -7, 10, -5, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 6 3/2 5/2 7/2 9/2 -q + ---- - ------- + 9 Sqrt[q] - 13 q + 14 q - 15 q + 13 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 17/2 > 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -4 2 2 4 6 8 10 12 18 20 -1 + q - q + -- + q + 2 q - q + 5 q - 2 q + 3 q + 2 q - 2 q + 2 q 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 377]][a, z]
Out[8]=	3 3 3 5 5 5 1 1 4 z 8 z 5 z 8 z 20 z 8 z 5 z 18 z 5 z -(----) + --- + --- - --- + --- + ---- - ----- + ---- + ---- - ----- + ---- + 3 a z 5 3 a 5 3 a 5 3 a a z a a a a a a 7 7 7 9 z 7 z z z > -- - ---- + -- - -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 377]][a, z]
Out[9]=	2 2 -2 1 1 z z 4 z 9 z 4 z 2 z 2 z -a + ---- + --- - -- - -- - --- - --- - --- + a z + 4 z + --- - ---- + 3 a z 9 7 5 3 a 10 8 a z a a a a a a 2 2 3 3 3 3 3 4 6 z 7 z 3 z z 11 z 27 z 8 z 3 4 z > ---- + ---- + ---- - -- + ----- + ----- + ---- - 4 a z - 13 z - --- + 4 2 9 7 5 3 a 10 a a a a a a a 4 4 4 4 5 5 5 5 5 5 z 3 z 16 z 20 z 3 z 6 z 11 z 40 z 16 z > ---- - ---- - ----- - ----- - ---- + ---- - ----- - ----- - ----- + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 5 6 5 z 8 z 16 z 15 z 6 z 10 z 32 z > 4 a z + 12 z - ---- + ---- + ----- + ----- - ---- + ----- + ----- + 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 10 15 z 7 8 6 z 2 z z 5 z 9 z 4 z 2 z 2 z > ----- - a z - 3 z - ---- - ---- + -- - ---- - ---- - ---- - ----- - ----- a 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 2 4 5 4 q 4 6 8 q + 6 q + ----- + ----- + ----- + -- + ----- + - + ---- + 7 q t + 7 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 12 5 > 8 q t + 8 q t + 6 q t + 7 q t + 4 q t + 6 q t + 2 q t + 14 5 14 6 16 6 18 7 > 4 q t + q t + 2 q t + q t

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