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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,6,15,5 X22,14,9,13 X18,8,19,7 X20,18,21,17 X16,22,17,21 X6,16,7,15 X8,20,1,19

Gauss Code:

{{1, -2, 3, -4, 5, -10, 7, -11}, {4, -1, 2, -3, 6, -5, 10, -9, 8, -7, 11, -8, 9, -6}}

Jones Polynomial:

- q-7/2 + 3q-5/2 - 7q-3/2 + 11q-1/2 - 15q1/2 + 16q3/2 - 17q5/2 + 14q7/2 - 10q9/2 + 6q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - q-8 + 2q-6 + q-4 - q-2 + 4 - 2q2 + 3q4 + q6 + 3q10 - 3q12 + 2q14 - q18 + q20 - q22

HOMFLY-PT Polynomial:

2a-5z + 3a-5z3 + a-5z5 - a-3z - 5a-3z3 - 4a-3z5 - a-3z7 - a-1z-1 - 4a-1z - 6a-1z3 - 4a-1z5 - a-1z7 + az-1 + 3az + 3az3 + az5

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + a-7z - 7a-7z3 + 9a-7z5 - 3a-7z7 + 2a-6z2 - 7a-6z4 + 10a-6z6 - 4a-6z8 - a-5z - a-5z3 + 3a-5z5 + 3a-5z7 - 3a-5z9 + 3a-4z2 - 12a-4z4 + 16a-4z6 - 6a-4z8 - a-4z10 + a-3z3 - 3a-3z5 + 8a-3z7 - 6a-3z9 - 7a-2z4 + 13a-2z6 - 7a-2z8 - a-2z10 - a-1z-1 + 6a-1z - 12a-1z3 + 12a-1z5 - 3a-1z7 - 3a-1z9 + 1 - z2 + 5z6 - 5z8 - az-1 + 3az - 5az3 + 8az5 - 5az7 - 2a2z2 + 5a2z4 - 3a2z6 - a3z + 2a3z3 - a3z5

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 r = 7 j = 16                       1 j = 14                     2   j = 12                   4 1   j = 10                 6 2     j = 8               8 4       j = 6             9 6         j = 4           7 8           j = 2         8 9             j = 0       5 9               j = -2     2 6                 j = -4   1 5                   j = -6   2                     j = -8 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[11, Alternating, 272]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 272]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[14, 6, 15, 5], X[22, 14, 9, 13], X[18, 8, 19, 7], X[20, 18, 21, 17], > X[16, 22, 17, 21], X[6, 16, 7, 15], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -10, 7, -11}, > {4, -1, 2, -3, 6, -5, 10, -9, 8, -7, 11, -8, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 7 11 3/2 5/2 7/2 -q + ---- - ---- + ------- - 15 Sqrt[q] + 16 q - 17 q + 14 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 10 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 2 -4 -2 2 4 6 10 12 14 4 + q - q + -- + q - q - 2 q + 3 q + q + 3 q - 3 q + 2 q - 6 q 18 20 22 > q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 272]][a, z]
Out[8]=	3 3 3 5 5 1 a 2 z z 4 z 3 z 5 z 6 z 3 z 4 z -(---) + - + --- - -- - --- + 3 a z + ---- - ---- - ---- + 3 a z + -- - ---- - a z z 5 3 a 5 3 a 5 3 a a a a a a 5 7 7 4 z 5 z z > ---- + a z - -- - -- a 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 272]][a, z]
Out[9]=	2 2 2 1 a z z 6 z 3 2 2 z 2 z 3 z 1 - --- - - + -- - -- + --- + 3 a z - a z - z - ---- + ---- + ---- - a z z 7 5 a 8 6 4 a a a a a 3 3 3 3 4 4 4 2 2 7 z z z 12 z 3 3 3 3 z 7 z 12 z > 2 a z - ---- - -- + -- - ----- - 5 a z + 2 a z + ---- - ---- - ----- - 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 6 7 z 2 4 9 z 3 z 3 z 12 z 5 3 5 6 z > ---- + 5 a z + ---- + ---- - ---- + ----- + 8 a z - a z + 5 z - -- + 2 7 5 3 a 8 a a a a a 6 6 6 7 7 7 7 10 z 16 z 13 z 2 6 3 z 3 z 8 z 3 z 7 > ----- + ----- + ----- - 3 a z - ---- + ---- + ---- - ---- - 5 a z - 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 8 4 z 6 z 7 z 3 z 6 z 3 z z z > 5 z - ---- - ---- - ---- - ---- - ---- - ---- - --- - --- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 2 5 6 2 4 9 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 9 q t + 7 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 2 6 2 6 3 8 3 8 4 10 4 10 5 > 8 q t + 9 q t + 6 q t + 8 q t + 4 q t + 6 q t + 2 q t + 12 5 12 6 14 6 16 7 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a272

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