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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,12,15,11 X20,15,5,16 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-24 - q-22 - 3q-18 + 2q-14 - q-12 + 4q-10 + 2q-8 + 3q-6 + 3q-4 - q-2 + 3 - 2q2 + 2q6 - q8

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-2z4 + 3a-1z3 - 4a-1z5 - z2 + 8z4 - 7z6 + az-1 - az - 2az3 + 8az5 - 7az7 - a2 - 2a2z2 + 8a2z4 - a2z6 - 4a2z8 + 2a3z - 13a3z3 + 21a3z5 - 9a3z7 - a3z9 + 3a4 - 7a4z2 + 2a4z4 + 9a4z6 - 6a4z8 - 2a5z-1 + 5a5z - 13a5z3 + 15a5z5 - 4a5z7 - a5z9 + 5a6 - 11a6z2 + 7a6z4 + 2a6z6 - 2a6z8 - a7z-1 + 2a7z - 5a7z3 + 6a7z5 - 2a7z7 + 2a8 - 5a8z2 + 4a8z4 - 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                 4 1   j = 0               6 3     j = -2             7 6       j = -4           6 4         j = -6         4 7           j = -8       4 6             j = -10     1 4               j = -12   1 4                 j = -14   1                   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, 36]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 36]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 12, 15, 11], X[20, 15, 5, 16], > X[16, 7, 17, 8], X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 5, -6, 7, -8, 3, -2, 10, -3, 4, -5, 8, -7, > 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 2 5 8 10 13 11 10 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 7 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 -22 3 2 -12 4 2 3 3 -2 2 6 8 3 - q - q - --- + --- - q + --- + -- + -- + -- - q - 2 q + 2 q - q 18 14 10 8 6 4 q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 36]][a, z]
Out[8]=	5 7 3 a 2 a a 3 5 7 z 3 3 3 5 3 -(-) + ---- - -- - a z - a z + 3 a z - a z + -- - a z - a z + 2 a z - z z z a 5 3 5 > a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 36]][a, z]
Out[9]=	5 7 2 4 6 8 a 2 a a 3 5 7 -a + 3 a + 5 a + 2 a + - - ---- - -- - a z + 2 a z + 5 a z + 2 a z - z z z 3 2 2 2 4 2 6 2 8 2 3 z 3 3 3 > z - 2 a z - 7 a z - 11 a z - 5 a z + ---- - 2 a z - 13 a z - a 4 5 3 7 3 4 z 2 4 4 4 6 4 8 4 > 13 a z - 5 a z + 8 z - -- + 8 a z + 2 a z + 7 a z + 4 a z - 2 a 5 4 z 5 3 5 5 5 7 5 6 2 6 4 6 > ---- + 8 a z + 21 a z + 15 a z + 6 a z - 7 z - a z + 9 a z + a 6 6 8 6 7 3 7 5 7 7 7 2 8 > 2 a z - a z - 7 a z - 9 a z - 4 a z - 2 a z - 4 a z - 4 8 6 8 3 9 5 9 > 6 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 1 4 1 4 4 6 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 4 7 6 4 7 2 2 2 4 2 6 3 > ----- + ----- + ----- + ---- + ---- + 3 t + 4 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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