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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X18,8,19,7 X22,20,5,19 X20,16,21,15 X16,22,17,21 X14,10,15,9 X10,14,11,13 X8,18,9,17 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 6q-3/2 + 9q-1/2 - 15q1/2 + 17q3/2 - 17q5/2 + 16q7/2 - 12q9/2 + 8q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 4q-6 + 2q-4 + 6 - q2 + q4 - 3q8 + q10 - 5q12 + q14 + q16 - 2q18 + 3q20 - q24

HOMFLY-PT Polynomial:

a-7z - 3a-5z - 3a-5z3 + a-3z-1 + 4a-3z + 4a-3z3 + 2a-3z5 - 2a-1z-1 - 3a-1z - a-1z3 + a-1z5 - 2az - 2az3 + a3z-1 + a3z

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + a-7z - 6a-7z3 + 10a-7z5 - 4a-7z7 + 2a-6z2 - 7a-6z4 + 14a-6z6 - 6a-6z8 + 2a-5z - 14a-5z3 + 17a-5z5 + a-5z7 - 4a-5z9 + 2a-4 + 7a-4z2 - 28a-4z4 + 33a-4z6 - 11a-4z8 - a-4z10 - a-3z-1 + a-3z - 7a-3z3 + 3a-3z5 + 9a-3z7 - 7a-3z9 + 5a-2 - 4a-2z2 - 14a-2z4 + 19a-2z6 - 8a-2z8 - a-2z10 - 2a-1z-1 + 3a-1z - 2a-1z3 + a-1z7 - 3a-1z9 + 3 - 8z2 + 8z4 - z6 - 3z8 + 3az5 - 3az7 - a2 + 3a2z4 - 2a2z6 + a3z-1 - 3a3z + 3a3z3 - 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                     3   j = 12                   5 1   j = 10                 7 3     j = 8               9 5       j = 6             8 7         j = 4           9 9           j = 2         6 8             j = 0       4 10               j = -2     2 5                 j = -4     4                   j = -6 1 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, 84]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 84]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[22, 20, 5, 19], > X[20, 16, 21, 15], X[16, 22, 17, 21], X[14, 10, 15, 9], X[10, 14, 11, 13], > X[8, 18, 9, 17], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 11, -2, 8, -7, 5, -6, 9, -3, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 6 9 3/2 5/2 7/2 -q + ---- - ---- + ------- - 15 Sqrt[q] + 17 q - 17 q + 16 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 12 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 4 2 2 4 8 10 12 14 16 18 6 + q + q + -- + -- - q + q - 3 q + q - 5 q + q + q - 2 q + 6 4 q q 20 24 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 84]][a, z]
Out[8]=	3 3 3 3 1 2 a z 3 z 4 z 3 z 3 3 z 4 z z ---- - --- + -- + -- - --- + --- - --- - 2 a z + a z - ---- + ---- - -- - 3 a z z 7 5 3 a 5 3 a a z a a a a a 5 5 3 2 z z > 2 a z + ---- + -- 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 84]][a, z]
Out[9]=	3 2 2 5 2 1 2 a z 2 z z 3 z 3 2 z 3 + -- + -- - a - ---- - --- + -- + -- + --- + -- + --- - 3 a z - 8 z - -- + 4 2 3 a z z 7 5 3 a 8 a a a z a a a a 2 2 2 3 3 3 3 4 2 z 7 z 4 z 6 z 14 z 7 z 2 z 3 3 4 2 z > ---- + ---- - ---- - ---- - ----- - ---- - ---- + 3 a z + 8 z + ---- - 6 4 2 7 5 3 a 8 a a a a a a a 4 4 4 5 5 5 7 z 28 z 14 z 2 4 10 z 17 z 3 z 5 3 5 > ---- - ----- - ----- + 3 a z + ----- + ----- + ---- + 3 a z - a z - 6 4 2 7 5 3 a a a a a a 6 6 6 6 7 7 7 7 6 z 14 z 33 z 19 z 2 6 4 z z 9 z z > z - -- + ----- + ----- + ----- - 2 a z - ---- + -- + ---- + -- - 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 6 z 11 z 8 z 4 z 7 z 3 z z z > 3 a z - 3 z - ---- - ----- - ---- - ---- - ---- - ---- - --- - --- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 4 2 4 5 2 10 + 6 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 8 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 9 q t + 9 q t + 8 q t + 7 q t + 9 q t + 5 q t + 7 q t + 10 5 12 5 12 6 14 6 16 7 > 3 q t + 5 q t + q t + 3 q t + q t

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