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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 2q-5/2 - 5q-3/2 + 8q-1/2 - 11q1/2 + 12q3/2 - 12q5/2 + 9q7/2 - 7q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

q-12 + q-10 + 3q-6 - 2q-2 + 2 - 2q2 + 2q4 + q6 + q8 + 3q10 - 2q12 + 2q14 - 2q18 + q20

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

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

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