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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-14 + q-12 + 2q-10 + 4q-6 - q-4 + 1 - 3q2 + 3q4 - 2q6 + 3q8 + 2q10 - q12 + 2q14 - q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-6z4 + 3a-5z3 - 4a-5z5 - 2a-4z2 + 7a-4z4 - 7a-4z6 + a-3z-1 - a-3z - a-3z3 + 8a-3z5 - 8a-3z7 - 6a-2z2 + 9a-2z4 + 2a-2z6 - 6a-2z8 + 2a-1z-1 - 2a-1z - 17a-1z3 + 30a-1z5 - 11a-1z7 - 2a-1z9 + 1 - 5z2 - 4z4 + 21z6 - 11z8 + 2az-1 - 21az3 + 29az5 - 7az7 - 2az9 - 2a2z2 - 3a2z4 + 11a2z6 - 5a2z8 + a3z-1 + a3z - 8a3z3 + 11a3z5 - 4a3z7 - a4z2 + 2a4z4 - 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                 4 1   j = 6               7 3     j = 4             7 4       j = 2           8 7         j = 0         7 9           j = -2       4 6             j = -4     3 7               j = -6   1 4                 j = -8   3                   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, 4]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 4]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[16, 10, 17, 9], X[20, 18, 5, 17], X[18, 11, 19, 12], > X[10, 19, 11, 20], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {4, -1, 2, -5, 6, -9, 8, -4, 10, -2, 3, -6, 7, -8, > 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 7 11 13 3/2 5/2 q - ---- + ---- - ---- + ------- - 15 Sqrt[q] + 14 q - 11 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 > 7 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 4 -4 2 4 6 8 10 12 1 - q + q + --- + -- - q - 3 q + 3 q - 2 q + 3 q + 2 q - q + 10 6 q q 14 16 > 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 4]][a, z]
Out[8]=	3 3 3 5 1 2 2 a a 3 2 z 5 z 3 3 3 z -(----) + --- - --- + -- + a z - a z + ---- - ---- + 5 a z - a z + -- - 3 a z z z 3 a 3 a z a a 5 7 4 z 5 z > ---- + 2 a z - -- a a
In[9]:=	Kauffman[Link[10, Alternating, 4]][a, z]
Out[9]=	3 2 2 1 2 2 a a z 2 z 3 2 2 z 6 z 2 2 1 + ---- + --- + --- + -- - -- - --- + a z - 5 z - ---- - ---- - 2 a z - 3 a z z z 3 a 4 2 a z a a a 3 3 3 4 4 4 4 2 3 z z 17 z 3 3 3 4 z 7 z 9 z > a z + ---- - -- - ----- - 21 a z - 8 a z - 4 z - -- + ---- + ---- - 5 3 a 6 4 2 a a a a a 5 5 5 2 4 4 4 4 z 8 z 30 z 5 3 5 6 > 3 a z + 2 a z - ---- + ---- + ----- + 29 a z + 11 a z + 21 z - 5 3 a a a 6 6 7 7 7 z 2 z 2 6 4 6 8 z 11 z 7 3 7 8 > ---- + ---- + 11 a z - a z - ---- - ----- - 7 a z - 4 a z - 11 z - 4 2 3 a a a a 8 9 6 z 2 8 2 z 9 > ---- - 5 a z - ---- - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 4 3 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 + 4 q t + 7 q t + 3 q t + 4 q t + q t + 10 4 12 5 > 3 q t + q t

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