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Knotscape

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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X20,18,7,17 X18,12,19,11 X12,20,13,19

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 5q-7/2 - 8q-5/2 + 9q-3/2 - 10q-1/2 + 8q1/2 - 7q3/2 + 4q5/2 - 2q7/2 + q9/2

A2 (sl(3)) Invariant:

- q-16 + q-14 - q-12 + q-10 + 2q-8 + 4q-4 + 3 + q2 + 2q6 - q8 - q12 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 3a-3z + a-3z3 - 3a-1z-1 - 9a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 7az + 9az3 + 5az5 + az7 - 2a3z - 3a3z3 - a3z5

Kauffman Polynomial:

a-4 - 4a-4z2 + 4a-4z4 - a-4z6 - a-3z-1 + 4a-3z - 7a-3z3 + 7a-3z5 - 2a-3z7 + 3a-2 - 7a-2z2 + 3a-2z4 + 4a-2z6 - 2a-2z8 - 3a-1z-1 + 12a-1z - 19a-1z3 + 14a-1z5 - 2a-1z7 - a-1z9 + 3 - 3z2 - 7z4 + 12z6 - 5z8 - 2az-1 + 9az - 17az3 + 14az5 - 4az7 - az9 - a2z2 - a2z4 + 3a2z6 - 3a2z8 - a3z3 + 4a3z5 - 4a3z7 + 4a4z4 - 4a4z6 - a5z + 4a5z3 - 3a5z5 + a6z2 - a6z4

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 = 10                     1 j = 8                   1   j = 6                 3 1   j = 4               4 1     j = 2             4 3       j = 0           6 4         j = -2         4 5           j = -4       4 5             j = -6     2 5               j = -8   1 3                 j = -10   2                   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, Alternating, 68]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 68]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[6, 7, 1, 8], > X[16, 13, 17, 14], X[14, 6, 15, 5], X[4, 16, 5, 15], X[20, 18, 7, 17], > X[18, 12, 19, 11], X[12, 20, 13, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -4}, > {4, -1, 2, -3, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 5 8 9 10 3/2 5/2 q - ---- + ---- - ---- + ---- - ------- + 8 Sqrt[q] - 7 q + 4 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 7/2 9/2 > 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 -10 2 4 2 6 8 12 14 3 - q + q - q + q + -- + -- + q + 2 q - q - q - q 8 4 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 68]][a, z]
Out[8]=	3 3 1 3 2 a 3 z 9 z 3 z 8 z 3 3 3 ---- - --- + --- + --- - --- + 7 a z - 2 a z + -- - ---- + 9 a z - 3 a z - 3 a z z 3 a 3 a a z a a 5 2 z 5 3 5 7 > ---- + 5 a z - a z + a z a
In[9]:=	Kauffman[Link[10, Alternating, 68]][a, z]
Out[9]=	2 -4 3 1 3 2 a 4 z 12 z 5 2 4 z 3 + a + -- - ---- - --- - --- + --- + ---- + 9 a z - a z - 3 z - ---- - 2 3 a z z 3 a 4 a a z a a 2 3 3 7 z 2 2 6 2 7 z 19 z 3 3 3 5 3 4 > ---- - a z + a z - ---- - ----- - 17 a z - a z + 4 a z - 7 z + 2 3 a a a 4 4 5 5 4 z 3 z 2 4 4 4 6 4 7 z 14 z 5 3 5 > ---- + ---- - a z + 4 a z - a z + ---- + ----- + 14 a z + 4 a z - 4 2 3 a a a a 6 6 7 7 5 5 6 z 4 z 2 6 4 6 2 z 2 z 7 > 3 a z + 12 z - -- + ---- + 3 a z - 4 a z - ---- - ---- - 4 a z - 4 2 3 a a a a 8 9 3 7 8 2 z 2 8 z 9 > 4 a z - 5 z - ---- - 3 a z - -- - a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 3 2 5 4 5 6 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 4 2 2 2 4 2 4 3 6 3 6 4 8 4 > ---- + 4 t + 4 q t + 3 q t + 4 q t + q t + 3 q t + q t + q t + 2 q t 10 5 > q t

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