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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 5q-11/2 - 8q-9/2 + 11q-7/2 - 12q-5/2 + 11q-3/2 - 10q-1/2 + 6q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

- q-24 + 2q-20 - q-18 + q-16 + q-14 - 3q-12 + q-10 - q-8 + q-6 + 2q-4 + 5 + q4 + 2q6 - q8

HOMFLY-PT Polynomial:

- a-1z-1 + a-1z3 + az-1 - az3 - az5 - a3z - a3z3 - a3z5 + 2a5z + 2a5z3 - a7z

Kauffman Polynomial:

- a-2z4 - a-1z-1 + 4a-1z3 - 4a-1z5 + 1 + 6z4 - 6z6 - az-1 + 2az3 + 5az5 - 6az7 + 4a2z2 - 6a2z4 + 7a2z6 - 5a2z8 + 2a3z - 11a3z3 + 13a3z5 - 3a3z7 - 2a3z9 + 9a4z2 - 28a4z4 + 27a4z6 - 9a4z8 + 4a5z - 17a5z3 + 14a5z5 - 2a5z9 + 4a6z2 - 12a6z4 + 13a6z6 - 4a6z8 + 2a7z - 8a7z3 + 10a7z5 - 3a7z7 - a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                     1 j = 4                   3   j = 2                 3 1   j = 0               7 3     j = -2             6 5       j = -4           6 5         j = -6         5 6           j = -8       3 6             j = -10     2 5               j = -12   1 3                 j = -14   2                   j = -16 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, 3]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 3]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[14, 10, 15, 9], > X[8, 4, 9, 3], X[10, 5, 11, 6], X[20, 11, 5, 12], X[18, 13, 19, 14], > X[12, 19, 13, 20], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 7, -9, 8, -4, 10, -2, 3, -8, > 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 8 11 12 11 10 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 6 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 > 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -18 -16 -14 3 -10 -8 -6 2 4 6 5 - q + --- - q + q + q - --- + q - q + q + -- + q + 2 q - 20 12 4 q q q 8 > q
In[8]:=	HOMFLYPT[Link[10, Alternating, 3]][a, z]
Out[8]=	3 1 a 3 5 7 z 3 3 3 5 3 5 3 5 -(---) + - - a z + 2 a z - a z + -- - a z - a z + 2 a z - a z - a z a z z a
In[9]:=	Kauffman[Link[10, Alternating, 3]][a, z]
Out[9]=	1 a 3 5 7 2 2 4 2 6 2 8 2 1 - --- - - + 2 a z + 4 a z + 2 a z + 4 a z + 9 a z + 4 a z - a z + a z z 3 4 4 z 3 3 3 5 3 7 3 4 z 2 4 > ---- + 2 a z - 11 a z - 17 a z - 8 a z + 6 z - -- - 6 a z - a 2 a 5 4 4 6 4 8 4 4 z 5 3 5 5 5 > 28 a z - 12 a z + 3 a z - ---- + 5 a z + 13 a z + 14 a z + a 7 5 6 2 6 4 6 6 6 8 6 7 > 10 a z - 6 z + 7 a z + 27 a z + 13 a z - a z - 6 a z - 3 7 7 7 2 8 4 8 6 8 3 9 5 9 > 3 a z - 3 a z - 5 a z - 9 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 3 2 5 3 6 7 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 5 6 6 5 6 2 2 2 4 2 6 3 > ----- + ----- + ----- + ---- + ---- + 3 t + 3 q t + q t + 3 q t + q t 6 3 6 2 4 2 4 2 q t q t q t q t q t

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