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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 13q-9/2 + 18q-7/2 - 22q-5/2 + 22q-3/2 - 20q-1/2 + 15q1/2 - 10q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + q-16 + 5q-14 - 2q-12 + 4q-10 - q-8 - 2q-6 + 2q-4 - 4q-2 + 6 - q2 + q4 + 4q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z-1 - 3a-1z - 2a-1z3 - a-1z5 + 5az-1 + 10az + 9az3 + 4az5 + az7 - 6a3z-1 - 12a3z - 9a3z3 - 3a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 4a-2z4 - 4a-2z6 - 2a-1z-1 + 7a-1z - 12a-1z3 + 15a-1z5 - 9a-1z7 + 1 - 4z4 + 13z6 - 10z8 - 5az-1 + 24az - 47az3 + 45az5 - 12az7 - 5az9 + a2 - 19a2z4 + 38a2z6 - 19a2z8 - a2z10 - 6a3z-1 + 31a3z - 60a3z3 + 50a3z5 - 6a3z7 - 8a3z9 + 3a4 - 6a4z2 - 10a4z4 + 27a4z6 - 13a4z8 - a4z10 - 4a5z-1 + 18a5z - 34a5z3 + 29a5z5 - 6a5z7 - 3a5z9 + 3a6 - 8a6z2 + 4a6z4 + 5a6z6 - 4a6z8 - a7z-1 + 4a7z - 8a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   7 1   j = 2                 8 3     j = 0               12 7       j = -2             12 10         j = -4           10 10           j = -6         8 12             j = -8       5 10               j = -10     2 8                 j = -12   1 5                   j = -14   2                     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, Alternating, 91]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 91]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[18, 8, 19, 7], X[22, 20, 5, 19], > X[20, 9, 21, 10], X[8, 21, 9, 22], X[16, 11, 17, 12], X[14, 17, 15, 18], > X[10, 15, 11, 16], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -2, 11, -8, 9, -7, 8, -3, > 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 13 18 22 22 20 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 10 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 -16 5 2 4 -8 2 2 4 6 - q - q + q - --- + q + --- - --- + --- - q - -- + -- - -- - 18 14 12 10 6 4 2 q q q q q q q 2 4 6 8 10 > q + q + 4 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 91]][a, z]
Out[8]=	3 5 7 3 -2 5 a 6 a 4 a a 3 z 3 5 7 2 z --- + --- - ---- + ---- - -- - --- + 10 a z - 12 a z + 6 a z - a z - ---- + a z z z z z a a 5 3 3 3 5 3 z 5 3 5 7 > 9 a z - 9 a z + 3 a z - -- + 4 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 91]][a, z]
Out[9]=	3 5 7 2 4 6 8 2 5 a 6 a 4 a a 7 z 1 + a + 3 a + 3 a + a - --- - --- - ---- - ---- - -- + --- + 24 a z + a z z z z z a 2 3 3 5 7 z 4 2 6 2 8 2 z > 31 a z + 18 a z + 4 a z - -- - 6 a z - 8 a z - 3 a z + -- - 2 3 a a 3 4 12 z 3 3 3 5 3 7 3 4 4 z 2 4 > ----- - 47 a z - 60 a z - 34 a z - 8 a z - 4 z + ---- - 19 a z - a 2 a 5 5 4 4 6 4 8 4 z 15 z 5 3 5 5 5 > 10 a z + 4 a z + 3 a z - -- + ----- + 45 a z + 50 a z + 29 a z + 3 a a 6 7 7 5 6 4 z 2 6 4 6 6 6 8 6 9 z > 8 a z + 13 z - ---- + 38 a z + 27 a z + 5 a z - a z - ---- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 12 a z - 6 a z - 6 a z - 3 a z - 10 z - 19 a z - 13 a z - 6 8 9 3 9 5 9 2 10 4 10 > 4 a z - 5 a z - 8 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 2 8 5 10 12 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 8 12 10 10 12 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 7 t + 8 q t + 3 q t + 7 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

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