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DrawMorseLink

PD Presentation:

X8192 X16,6,17,5 X18,10,19,9 X10,20,11,19 X2,11,3,12 X12,3,13,4 X4758 X20,16,21,15 X22,14,7,13 X14,22,15,21 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 7q-3/2 + 11q-1/2 - 16q1/2 + 17q3/2 - 18q5/2 + 15q7/2 - 11q9/2 + 7q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - q-8 + 2q-6 + q-4 - q-2 + 5 - q2 + 4q4 + 2q6 + 3q10 - 4q12 + q14 - q16 - 2q18 + q20 - q22

HOMFLY-PT Polynomial:

a-5z-1 + 3a-5z + 3a-5z3 + a-5z5 - 2a-3z-1 - 4a-3z - 6a-3z3 - 4a-3z5 - a-3z7 - 3a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az-1 + 3az + 3az3 + az5

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 - 4a-7z3 + 8a-7z5 - 3a-7z7 - 2a-6 + 9a-6z2 - 13a-6z4 + 14a-6z6 - 5a-6z8 + a-5z-1 - 4a-5z + 12a-5z3 - 14a-5z5 + 13a-5z7 - 5a-5z9 - 5a-4 + 22a-4z2 - 35a-4z4 + 24a-4z6 - 4a-4z8 - 2a-4z10 + 2a-3z-1 - 5a-3z + 14a-3z3 - 32a-3z5 + 28a-3z7 - 10a-3z9 - 3a-2 + 13a-2z2 - 30a-2z4 + 22a-2z6 - 5a-2z8 - 2a-2z10 + 4a-1z - 8a-1z3 - a-1z5 + 7a-1z7 - 5a-1z9 + 1 + z2 - 6z4 + 10z6 - 6z8 - az-1 + 4az - 4az3 + 8az5 - 5az7 - a2z2 + 5a2z4 - 3a2z6 - a3z + 2a3z3 - a3z5

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 = 16                       1 j = 14                     2   j = 12                   5 1   j = 10                 6 2     j = 8               9 5       j = 6             9 6         j = 4           8 9           j = 2         8 9             j = 0       4 9               j = -2     3 7                 j = -4   1 5                   j = -6   2                     j = -8 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, 225]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 225]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 6, 17, 5], X[18, 10, 19, 9], X[10, 20, 11, 19], > X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 7, 5, 8], X[20, 16, 21, 15], > X[22, 14, 7, 13], X[14, 22, 15, 21], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -5, 6, -7, 2, -11}, > {7, -1, 3, -4, 5, -6, 9, -10, 8, -2, 11, -3, 4, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 7 11 3/2 5/2 7/2 -q + ---- - ---- + ------- - 16 Sqrt[q] + 17 q - 18 q + 15 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 11 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 2 -4 -2 2 4 6 10 12 14 5 + q - q + -- + q - q - q + 4 q + 2 q + 3 q - 4 q + q - 6 q 16 18 20 22 > q - 2 q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 225]][a, z]
Out[8]=	3 3 3 5 1 2 a 3 z 4 z 3 z 3 z 6 z 6 z 3 z ---- - ---- + - + --- - --- - --- + 3 a z + ---- - ---- - ---- + 3 a z + -- - 5 3 z 5 3 a 5 3 a 5 a z a z a a a a a 5 5 7 7 4 z 4 z 5 z z > ---- - ---- + a z - -- - -- 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 225]][a, z]
Out[9]=	2 5 3 1 2 a 4 z 5 z 4 z 3 2 1 - -- - -- - -- + ---- + ---- - - - --- - --- + --- + 4 a z - a z + z - 6 4 2 5 3 z 5 3 a a a a a z a z a a 2 2 2 2 3 3 3 3 2 z 9 z 22 z 13 z 2 2 4 z 12 z 14 z 8 z > ---- + ---- + ----- + ----- - a z - ---- + ----- + ----- - ---- - 8 6 4 2 7 5 3 a a a a a a a a 4 4 4 4 5 3 3 3 4 3 z 13 z 35 z 30 z 2 4 8 z > 4 a z + 2 a z - 6 z + ---- - ----- - ----- - ----- + 5 a z + ---- - 8 6 4 2 7 a a a a a 5 5 5 6 6 6 6 14 z 32 z z 5 3 5 6 z 14 z 24 z 22 z > ----- - ----- - -- + 8 a z - a z + 10 z - -- + ----- + ----- + ----- - 5 3 a 8 6 4 2 a a a a a a 7 7 7 7 8 8 2 6 3 z 13 z 28 z 7 z 7 8 5 z 4 z > 3 a z - ---- + ----- + ----- + ---- - 5 a z - 6 z - ---- - ---- - 7 5 3 a 6 4 a a a a a 8 9 9 9 10 10 5 z 5 z 10 z 5 z 2 z 2 z > ---- - ---- - ----- - ---- - ----- - ----- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 3 4 7 2 4 9 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 9 q t + 8 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 9 q t + 9 q t + 6 q t + 9 q t + 5 q t + 6 q t + 2 q t + 12 5 12 6 14 6 16 7 > 5 q t + q t + 2 q t + q t

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