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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 X16,11,17,12 X18,13,19,14 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, -7, 9, -8, 5, -3}}

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 9q-11/2 + 11q-9/2 - 12q-7/2 + 10q-5/2 - 9q-3/2 + 5q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-26 - 2q-20 + 2q-18 - q-16 + q-14 + 2q-12 + 5q-8 + 3q-4 + q-2 - 1 + q2 - q4

HOMFLY-PT Polynomial:

2az + 3az3 + az5 - 2a3z-1 - 9a3z - 10a3z3 - 5a3z5 - a3z7 + 3a5z-1 + 8a5z + 7a5z3 + 2a5z5 - a7z-1 - 2a7z - a7z3

Kauffman Polynomial:

- 2z2 + 3z4 - z6 + 4az - 10az3 + 10az5 - 3az7 - 3a2z2 + a2z4 + 6a2z6 - 3a2z8 - 2a3z-1 + 15a3z - 32a3z3 + 29a3z5 - 7a3z7 - a3z9 + 3a4 - 6a4z2 - 3a4z4 + 14a4z6 - 7a4z8 - 3a5z-1 + 16a5z - 35a5z3 + 32a5z5 - 10a5z7 - a5z9 + 3a6 - 9a6z2 + 6a6z4 + 2a6z6 - 4a6z8 - a7z-1 + 5a7z - 10a7z3 + 10a7z5 - 6a7z7 + a8 - 3a8z2 + 6a8z4 - 5a8z6 + 3a9z3 - 3a9z5 + a10z2 - a10z4

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 = 4                     1 j = 2                   2   j = 0                 3 1   j = -2               6 2     j = -4             5 4       j = -6           7 5         j = -8         5 6           j = -10       4 6             j = -12     2 5               j = -14   1 4                 j = -16   2                   j = -18 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, 63]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 63]]
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[16, 11, 17, 12], X[18, 13, 19, 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, -7, 9, -8, 5, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 9 11 12 10 9 5 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 > 3 Sqrt[q] + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 2 2 -16 -14 2 5 3 -2 2 4 -1 - q - --- + --- - q + q + --- + -- + -- + q + q - q 20 18 12 8 4 q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 63]][a, z]
Out[8]=	3 5 7 -2 a 3 a a 3 5 7 3 3 3 ----- + ---- - -- + 2 a z - 9 a z + 8 a z - 2 a z + 3 a z - 10 a z + z z z 5 3 7 3 5 3 5 5 5 3 7 > 7 a z - a z + a z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 63]][a, z]
Out[9]=	3 5 7 4 6 8 2 a 3 a a 3 5 7 3 a + 3 a + a - ---- - ---- - -- + 4 a z + 15 a z + 16 a z + 5 a z - z z z 2 2 2 4 2 6 2 8 2 10 2 3 > 2 z - 3 a z - 6 a z - 9 a z - 3 a z + a z - 10 a z - 3 3 5 3 7 3 9 3 4 2 4 4 4 > 32 a z - 35 a z - 10 a z + 3 a z + 3 z + a z - 3 a z + 6 4 8 4 10 4 5 3 5 5 5 7 5 > 6 a z + 6 a z - a z + 10 a z + 29 a z + 32 a z + 10 a z - 9 5 6 2 6 4 6 6 6 8 6 7 3 7 > 3 a z - z + 6 a z + 14 a z + 2 a z - 5 a z - 3 a z - 7 a z - 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 10 a z - 6 a z - 3 a z - 7 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 6 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 5 6 7 5 5 2 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + t + 2 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 4 3 > q t

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