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DrawMorseLink

PD Presentation:

X6172 X2,9,3,10 X12,3,13,4 X20,12,5,11 X18,6,19,5 X10,20,11,19 X4,17,1,18 X16,14,17,13 X14,8,15,7 X8,16,9,15

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 8q-3/2 + 11q-1/2 - 16q1/2 + 17q3/2 - 16q5/2 + 13q7/2 - 9q9/2 + 5q11/2 - q13/2

A2 (sl(3)) Invariant:

q-10 - q-8 + 4q-6 + 2q-4 + q-2 + 7 - q2 + 3q4 - 2q6 - 2q8 + q10 - 4q12 + 2q14 - q16 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-5z3 + 2a-3z-1 + 5a-3z + 5a-3z3 + 2a-3z5 - 5a-1z-1 - 12a-1z - 12a-1z3 - 5a-1z5 - a-1z7 + 3az-1 + 5az + 3az3 + az5

Kauffman Polynomial:

- a-7z5 - a-6 + 6a-6z4 - 5a-6z6 - 5a-5z3 + 15a-5z5 - 9a-5z7 - 4a-4z2 + 9a-4z4 + 4a-4z6 - 7a-4z8 - 2a-3z-1 + 5a-3z - 18a-3z3 + 34a-3z5 - 15a-3z7 - 2a-3z9 + 5a-2 - 17a-2z2 + 10a-2z4 + 13a-2z6 - 12a-2z8 - 5a-1z-1 + 16a-1z - 29a-1z3 + 31a-1z5 - 12a-1z7 - 2a-1z9 + 5 - 14z2 + 11z4 + z6 - 5z8 - 3az-1 + 10az - 14az3 + 12az5 - 6az7 - a2z2 + 4a2z4 - 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 j = 14                     1 j = 12                   4   j = 10                 5 1   j = 8               8 4     j = 6             8 5       j = 4           9 8         j = 2         7 8           j = 0       5 10             j = -2     3 6               j = -4     5                 j = -6 1 3                   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[10, Alternating, 43]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 43]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 9, 3, 10], X[12, 3, 13, 4], X[20, 12, 5, 11], > X[18, 6, 19, 5], X[10, 20, 11, 19], X[4, 17, 1, 18], X[16, 14, 17, 13], > X[14, 8, 15, 7], X[8, 16, 9, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7}, {5, -1, 9, -10, 2, -6, 4, -3, 8, -9, 10, -8, 7, -5, > 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 8 11 3/2 5/2 7/2 -q + ---- - ---- + ------- - 16 Sqrt[q] + 17 q - 16 q + 13 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 > 9 q + 5 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 4 2 -2 2 4 6 8 10 12 7 + q - q + -- + -- + q - q + 3 q - 2 q - 2 q + q - 4 q + 6 4 q q 14 16 18 20 > 2 q - q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 43]][a, z]
Out[8]=	3 3 3 5 2 5 3 a 5 z 12 z z 5 z 12 z 3 2 z ---- - --- + --- + --- - ---- + 5 a z - -- + ---- - ----- + 3 a z + ---- - 3 a z z 3 a 5 3 a 3 a z a a a a 5 7 5 z 5 z > ---- + a z - -- a a
In[9]:=	Kauffman[Link[10, Alternating, 43]][a, z]
Out[9]=	2 -6 5 2 5 3 a 5 z 16 z 3 2 4 z 5 - a + -- - ---- - --- - --- + --- + ---- + 10 a z - a z - 14 z - ---- - 2 3 a z z 3 a 4 a a z a a 2 3 3 3 4 17 z 2 2 5 z 18 z 29 z 3 3 3 4 6 z > ----- - a z - ---- - ----- - ----- - 14 a z + 2 a z + 11 z + ---- + 2 5 3 a 6 a a a a 4 4 5 5 5 5 9 z 10 z 2 4 z 15 z 34 z 31 z 5 3 5 > ---- + ----- + 4 a z - -- + ----- + ----- + ----- + 12 a z - a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 5 z 4 z 13 z 2 6 9 z 15 z 12 z 7 8 > z - ---- + ---- + ----- - 3 a z - ---- - ----- - ----- - 6 a z - 5 z - 6 4 2 5 3 a a a a a a 8 8 9 9 7 z 12 z 2 z 2 z > ---- - ----- - ---- - ---- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 3 5 3 5 6 2 10 + 7 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 + 8 q t + 8 q t + 5 q t + 8 q t + 4 q t + 5 q t + 10 5 12 5 14 6 > q t + 4 q t + q t

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