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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 5q-1/2 + 6q1/2 - 8q3/2 + 9q5/2 - 9q7/2 + 7q9/2 - 5q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-4 + 3q-2 + 1 + 2q2 - 2q6 + q8 - 2q10 + 2q12 + q18 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z-1 - 2a-5z - 3a-5z3 - a-5z5 + 3a-3z-1 + 8a-3z + 9a-3z3 + 5a-3z5 + a-3z7 - 4a-1z-1 - 9a-1z - 8a-1z3 - 2a-1z5 + 2az-1 + 3az + az3

Kauffman Polynomial:

- a-9z3 + a-8z2 - 3a-8z4 - a-7z + 4a-7z3 - 5a-7z5 + a-6 - 2a-6z2 + 7a-6z4 - 6a-6z6 - a-5z-1 + 2a-5z - 3a-5z3 + 10a-5z5 - 6a-5z7 + 3a-4 - 10a-4z2 + 8a-4z4 + 5a-4z6 - 4a-4z8 - 3a-3z-1 + 15a-3z - 34a-3z3 + 32a-3z5 - 7a-3z7 - a-3z9 + 3a-2 - 7a-2z2 - 10a-2z4 + 19a-2z6 - 6a-2z8 - 4a-1z-1 + 19a-1z - 35a-1z3 + 22a-1z5 - 2a-1z7 - a-1z9 + 2 - 8z4 + 8z6 - 2z8 - 2az-1 + 7az - 9az3 + 5az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                     1 j = 14                   2   j = 12                 3 1   j = 10               4 2     j = 8             5 3       j = 6           4 4         j = 4         4 5           j = 2       4 6             j = 0     1 2               j = -2   1 4                 j = -4   1                   j = -6 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, 27]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 27]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 8, 15, 7], X[16, 10, 17, 9], > X[8, 16, 9, 15], X[20, 18, 5, 17], X[18, 11, 19, 12], X[10, 19, 11, 20], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -5, 4, -8, 7, -2, 10, -3, 5, -4, 6, -7, > 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 5 3/2 5/2 7/2 9/2 -q + ---- - ------- + 6 Sqrt[q] - 8 q + 9 q - 9 q + 7 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 > 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -4 3 2 6 8 10 12 18 20 22 1 + q + q + q + -- + 2 q - 2 q + q - 2 q + 2 q + q - q + q 2 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 27]][a, z]
Out[8]=	3 3 3 1 3 4 2 a 2 z 8 z 9 z 3 z 9 z 8 z -(----) + ---- - --- + --- - --- + --- - --- + 3 a z - ---- + ---- - ---- + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 5 7 3 z 5 z 2 z z > a z - -- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[10, Alternating, 27]][a, z]
Out[9]=	-6 3 3 1 3 4 2 a z 2 z 15 z 19 z 2 + a + -- + -- - ---- - ---- - --- - --- - -- + --- + ---- + ---- + 7 a z + 4 2 5 3 a z z 7 5 3 a a a a z a z a a a 2 2 2 2 3 3 3 3 3 z 2 z 10 z 7 z z 4 z 3 z 34 z 35 z 3 > -- - ---- - ----- - ---- - -- + ---- - ---- - ----- - ----- - 9 a z - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 5 4 3 z 7 z 8 z 10 z 5 z 10 z 32 z 22 z 5 > 8 z - ---- + ---- + ---- - ----- - ---- + ----- + ----- + ----- + 5 a z + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 7 8 6 6 z 5 z 19 z 6 z 7 z 2 z 7 8 4 z > 8 z - ---- + ---- + ----- - ---- - ---- - ---- - a z - 2 z - ---- - 6 4 2 5 3 a 4 a a a a a a 8 9 9 6 z z z > ---- - -- - -- 2 3 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 4 2 4 q 4 6 q + 4 q + ----- + ----- + ----- + t + ----- + - + ---- + 5 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 4 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 3 q t + 12 5 14 5 16 6 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a27

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