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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 6q-1/2 + 6q1/2 - 11q3/2 + 9q5/2 - 11q7/2 + 8q9/2 - 6q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

q-8 + q-6 + 2q-4 + 6q-2 + 6 + 11q2 + 12q4 + 11q6 + 13q8 + 6q10 + 7q12 + 2q14 + q16 + 2q18 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z-3 - 2a-5z-1 - 3a-5z - 3a-5z3 - a-5z5 + 3a-3z-3 + 7a-3z-1 + 11a-3z + 10a-3z3 + 5a-3z5 + a-3z7 - 3a-1z-3 - 8a-1z-1 - 11a-1z - 8a-1z3 - 2a-1z5 + az-3 + 3az-1 + 3az + az3

Kauffman Polynomial:

- a-9z3 - 3a-8z4 - 2a-7z + 5a-7z3 - 6a-7z5 - 2a-6z2 + 10a-6z4 - 8a-6z6 + a-5z-3 - 5a-5z-1 + 12a-5z - 17a-5z3 + 20a-5z5 - 9a-5z7 - 3a-4z-2 + 10a-4 - 20a-4z2 + 17a-4z4 + 4a-4z6 - 5a-4z8 + 3a-3z-3 - 12a-3z-1 + 35a-3z - 61a-3z3 + 51a-3z5 - 12a-3z7 - a-3z9 - 6a-2z-2 + 19a-2 - 26a-2z2 + 19a-2z6 - 7a-2z8 + 3a-1z-3 - 12a-1z-1 + 31a-1z - 48a-1z3 + 30a-1z5 - 4a-1z7 - a-1z9 - 3z-2 + 10 - 8z2 - 4z4 + 7z6 - 2z8 + az-3 - 5az-1 + 10az - 10az3 + 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                   3 1 j = 12                 3     j = 10               5 3     j = 8             6 3       j = 6           5 7         j = 4         6 4           j = 2       4 9             j = 0     2 2               j = -2   1 5                 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 172]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 172]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[20, 15, 17, 16], X[14, 8, 15, 7], > X[10, 12, 5, 11], X[16, 19, 11, 20], X[8, 18, 9, 17], X[18, 10, 19, 9], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {7, -8, 6, -3}, {9, -1, 4, -7, 8, -5}, > {5, -2, 10, -4, 3, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 6 3/2 5/2 7/2 9/2 -q + ---- - ------- + 6 Sqrt[q] - 11 q + 9 q - 11 q + 8 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 2 6 2 4 6 8 10 12 6 + q + q + -- + -- + 11 q + 12 q + 11 q + 13 q + 6 q + 7 q + 4 2 q q 14 16 18 20 22 > 2 q + q + 2 q - q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 172]][a, z]
Out[8]=	1 3 3 a 2 7 8 3 a 3 z 11 z 11 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- - --- + ---- - ---- + 5 3 3 3 3 3 5 3 a z z 5 3 a a z a z a z z a z a z a a 3 3 3 5 5 5 7 3 z 10 z 8 z 3 z 5 z 2 z z > 3 a z - ---- + ----- - ---- + a z - -- + ---- - ---- + -- 5 3 a 5 3 a 3 a a a a a
In[9]:=	Kauffman[Link[10, Alternating, 172]][a, z]
Out[9]=	10 19 1 3 3 a 3 3 6 5 12 10 + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- - ---- - ---- - 4 2 5 3 3 3 3 3 2 4 2 2 2 5 3 a a a z a z a z z z a z a z a z a z 2 2 12 5 a 2 z 12 z 35 z 31 z 2 2 z 20 z > --- - --- - --- + ---- + ---- + ---- + 10 a z - 8 z - ---- - ----- - a z z 7 5 3 a 6 4 a a a a a 2 3 3 3 3 3 4 4 26 z z 5 z 17 z 61 z 48 z 3 4 3 z 10 z > ----- - -- + ---- - ----- - ----- - ----- - 10 a z - 4 z - ---- + ----- + 2 9 7 5 3 a 8 6 a a a a a a a 4 5 5 5 5 6 6 17 z 6 z 20 z 51 z 30 z 5 6 8 z 4 z > ----- - ---- + ----- + ----- + ----- + 5 a z + 7 z - ---- + ---- + 4 7 5 3 a 6 4 a a a a a a 6 7 7 7 8 8 9 9 19 z 9 z 12 z 4 z 7 8 5 z 7 z z z > ----- - ---- - ----- - ---- - a z - 2 z - ---- - ---- - -- - -- 2 5 3 a 4 2 3 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 2 5 2 4 q 4 6 9 q + 6 q + ----- + ----- + ----- + -- + ----- + - + ---- + 4 q t + 5 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 2 8 2 8 3 10 3 10 4 12 4 14 5 > 7 q t + 6 q t + 3 q t + 5 q t + 3 q t + 3 q t + 3 q t + 14 6 16 6 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a172

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