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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X7,16,8,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-11/2 + q-9/2 - q-7/2 + q-5/2 - q-1/2 - q3/2 + q5/2 - q7/2

A2 (sl(3)) Invariant:

q-18 + q-16 + q-14 + q-12 - 2q-6 - q-4 + 1 + 2q2 + q4 + q6 + q8 + q10 + q12

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z + a-1z-1 + 3a-1z + a-1z3 + az-1 - 2a3z-1 - 3a3z - a3z3 + a5z-1 + a5z

Kauffman Polynomial:

a-3z-1 - 3a-3z + 4a-3z3 - a-3z5 - a-2 - 2a-2z2 + 4a-2z4 - a-2z6 + a-1z-1 - 5a-1z + 5a-1z3 - a-1z5 - 2 + 3z2 - az-1 + 6az - 12az3 + 7az5 - az7 - 3a2 + 11a2z2 - 14a2z4 + 7a2z6 - a2z8 - 2a3z-1 + 13a3z - 23a3z3 + 13a3z5 - 2a3z7 - a4 + 6a4z2 - 10a4z4 + 6a4z6 - a4z8 - a5z-1 + 5a5z - 10a5z3 + 6a5z5 - a5z7

Khovanov Homology:

trqj 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                       j = 4                 1 1   j = 2             2 1       j = 0             2 1       j = -2         1 2 2         j = -4       1               j = -6       1 1             j = -8   1 1                 j = -10                       j = -12 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, NonAlternating, 32]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, NonAlternating, 32]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[7, 16, 8, 17], X[17, 20, 18, 5], > X[11, 19, 12, 18], X[19, 11, 20, 10], X[9, 14, 10, 15], X[15, 8, 16, 9], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, -3, 8, -7, 6, -5, -2, 10, 7, -8, 3, -4, 5, > -6, 4}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) -(9/2) -(7/2) -(5/2) 1 3/2 5/2 7/2 -q + q - q + q - ------- - q + q - q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 -14 -12 2 -4 2 4 6 8 10 12 1 + q + q + q + q - -- - q + 2 q + q + q + q + q + q 6 q
In[8]:=	HOMFLYPT[Link[10, NonAlternating, 32]][a, z]
Out[8]=	3 5 3 1 1 a 2 a a z 3 z 3 5 z 3 3 -(----) + --- + - - ---- + -- - -- + --- - 3 a z + a z + -- - a z 3 a z z z z 3 a a a z a
In[9]:=	Kauffman[Link[10, NonAlternating, 32]][a, z]
Out[9]=	3 5 -2 2 4 1 1 a 2 a a 3 z 5 z -2 - a - 3 a - a + ---- + --- - - - ---- - -- - --- - --- + 6 a z + 3 a z z z z 3 a a z a 2 3 3 3 5 2 2 z 2 2 4 2 4 z 5 z > 13 a z + 5 a z + 3 z - ---- + 11 a z + 6 a z + ---- + ---- - 2 3 a a a 4 5 5 3 3 3 5 3 4 z 2 4 4 4 z z > 12 a z - 23 a z - 10 a z + ---- - 14 a z - 10 a z - -- - -- + 2 3 a a a 6 5 3 5 5 5 z 2 6 4 6 7 3 7 > 7 a z + 13 a z + 6 a z - -- + 7 a z + 6 a z - a z - 2 a z - 2 a 5 7 2 8 4 8 > a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 1 1 1 1 1 1 1 2 2 + -- + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 6 8 5 8 4 6 3 4 3 6 2 2 2 2 q q t q t q t q t q t q t q t q t 2 4 2 4 3 8 4 > t + q t + q t + q t + q t

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