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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X20,10,7,9 X2738 X4,20,5,19 X14,5,15,6 X18,11,19,12 X16,13,17,14 X12,17,13,18 X6,15,1,16

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-24 + 2q-20 - q-18 + q-16 + q-14 - 2q-12 + 2q-10 + 2q-6 + 2q-4 + 4 - q2 + q6 - q8

HOMFLY-PT Polynomial:

a-1z + a-1z3 - az-1 - 3az - 2az3 - az5 + a3z-1 - a3z3 - a3z5 + 2a5z + 2a5z3 - a7z

Kauffman Polynomial:

a-2z2 - a-2z4 - 2a-1z + 4a-1z3 - 3a-1z5 + 3z4 - 4z6 + az-1 - 5az + 5az3 + az5 - 4az7 - a2 + 3a2z4 - 3a2z8 + a3z-1 - a3z - 7a3z3 + 14a3z5 - 6a3z7 - a3z9 + 2a4z2 - 6a4z4 + 12a4z6 - 6a4z8 + 4a5z - 17a5z3 + 20a5z5 - 5a5z7 - a5z9 - a6z2 - 2a6z4 + 7a6z6 - 3a6z8 + 2a7z - 9a7z3 + 10a7z5 - 3a7z7 - 2a8z2 + 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                   2   j = 2                 3 1   j = 0               6 2     j = -2             5 4       j = -4           6 5         j = -6         5 6           j = -8       3 5             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, 65]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 65]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[20, 10, 7, 9], X[2, 7, 3, 8], > X[4, 20, 5, 19], X[14, 5, 15, 6], X[18, 11, 19, 12], X[16, 13, 17, 14], > X[12, 17, 13, 18], X[6, 15, 1, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -5, 6, -10}, > {4, -1, 3, -2, 7, -9, 8, -6, 10, -8, 9, -7, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 5 8 10 11 10 9 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 5 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 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 2 -18 -16 -14 2 2 2 2 2 6 8 4 - q + --- - q + q + q - --- + --- + -- + -- - q + q - q 20 12 10 6 4 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 65]][a, z]
Out[8]=	3 3 a a z 5 7 z 3 3 3 5 3 5 -(-) + -- + - - 3 a z + 2 a z - a z + -- - 2 a z - a z + 2 a z - a z - z z a a 3 5 > a z
In[9]:=	Kauffman[Link[10, Alternating, 65]][a, z]
Out[9]=	3 2 2 a a 2 z 3 5 7 z 4 2 6 2 -a + - + -- - --- - 5 a z - a z + 4 a z + 2 a z + -- + 2 a z - a z - z z a 2 a 3 4 8 2 4 z 3 3 3 5 3 7 3 4 z > 2 a z + ---- + 5 a z - 7 a z - 17 a z - 9 a z + 3 z - -- + a 2 a 5 2 4 4 4 6 4 8 4 3 z 5 3 5 5 5 > 3 a z - 6 a z - 2 a z + 3 a z - ---- + a z + 14 a z + 20 a z + a 7 5 6 4 6 6 6 8 6 7 3 7 5 7 > 10 a z - 4 z + 12 a z + 7 a z - a z - 4 a z - 6 a z - 5 a z - 7 7 2 8 4 8 6 8 3 9 5 9 > 3 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 2 1 3 2 5 3 5 6 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 5 2 2 2 4 2 6 3 > ----- + ----- + ----- + ---- + ---- + 2 t + 3 q t + q t + 2 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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