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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X14,10,15,9 X8493 X12,5,13,6 X20,13,5,14 X16,12,17,11 X10,16,11,15 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 6q-5/2 - 11q-3/2 + 13q-1/2 - 15q1/2 + 14q3/2 - 12q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

- q-14 + q-10 - q-8 + 4q-6 + q-2 + 3 - 2q2 + 4q4 - 2q6 + 2q8 + q10 - 2q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z + 2a-3z3 + a-3z5 - a-1z-1 - 4a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az-1 + 5az + 6az3 + 2az5 - 2a3z - a3z3

Kauffman Polynomial:

- a-6z4 + 2a-5z3 - 4a-5z5 - 2a-4z2 + 8a-4z4 - 8a-4z6 + 2a-3z - 7a-3z3 + 14a-3z5 - 10a-3z7 - 3a-2z2 + 5a-2z4 + 5a-2z6 - 7a-2z8 - a-1z-1 + 8a-1z - 26a-1z3 + 34a-1z5 - 12a-1z7 - 2a-1z9 + 1 - 11z4 + 23z6 - 11z8 - az-1 + 10az - 26az3 + 25az5 - 5az7 - 2az9 - a2z2 - 4a2z4 + 9a2z6 - 4a2z8 + 4a3z - 9a3z3 + 9a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                     1 j = 10                   3   j = 8                 5 1   j = 6               7 3     j = 4             7 5       j = 2           8 7         j = 0         7 9           j = -2       4 6             j = -4     2 7               j = -6   1 4                 j = -8   2                   j = -10 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, 5]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 5]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[14, 10, 15, 9], > X[8, 4, 9, 3], X[12, 5, 13, 6], X[20, 13, 5, 14], X[16, 12, 17, 11], > X[10, 16, 11, 15], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {6, -1, 2, -5, 4, -9, 8, -6, 7, -4, 9, -8, 10, -2, > 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 11 13 3/2 5/2 q - ---- + ---- - ---- + ------- - 15 Sqrt[q] + 14 q - 12 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -10 -8 4 -2 2 4 6 8 10 12 3 - q + q - q + -- + q - 2 q + 4 q - 2 q + 2 q + q - 2 q + 6 q 14 16 > 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 5]][a, z]
Out[8]=	3 3 5 1 a z 4 z 3 2 z 6 z 3 3 3 z -(---) + - + -- - --- + 5 a z - 2 a z + ---- - ---- + 6 a z - a z + -- - a z z 3 a 3 a 3 a a a 5 7 4 z 5 z > ---- + 2 a z - -- a a
In[9]:=	Kauffman[Link[10, Alternating, 5]][a, z]
Out[9]=	2 2 1 a 2 z 8 z 3 2 z 3 z 2 2 4 2 1 - --- - - + --- + --- + 10 a z + 4 a z - ---- - ---- - a z - 2 a z + a z z 3 a 4 2 a a a 3 3 3 4 4 4 2 z 7 z 26 z 3 3 3 4 z 8 z 5 z > ---- - ---- - ----- - 26 a z - 9 a z - 11 z - -- + ---- + ---- - 5 3 a 6 4 2 a a a a a 5 5 5 2 4 4 4 4 z 14 z 34 z 5 3 5 6 > 4 a z + 3 a z - ---- + ----- + ----- + 25 a z + 9 a z + 23 z - 5 3 a a a 6 6 7 7 8 z 5 z 2 6 4 6 10 z 12 z 7 3 7 8 > ---- + ---- + 9 a z - a z - ----- - ----- - 5 a z - 3 a z - 11 z - 4 2 3 a a a a 8 9 7 z 2 8 2 z 9 > ---- - 4 a z - ---- - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 7 4 7 6 9 + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 7 q t + 7 q t + 5 q t + 7 q t + 3 q t + 5 q t + q t + 10 4 12 5 > 3 q t + q t

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