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DrawMorseLink

PD Presentation:

X6172 X18,7,19,8 X4,19,1,20 X14,6,15,5 X8493 X16,10,17,9 X10,16,11,15 X20,12,21,11 X22,14,5,13 X12,22,13,21 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-3/2 - 3q-1/2 + 5q1/2 - 10q3/2 + 13q5/2 - 17q7/2 + 16q9/2 - 15q11/2 + 12q13/2 - 7q15/2 + 4q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + q-2 + 3q4 - q6 + 6q8 + 3q10 + 3q12 + 4q14 - 3q16 + q18 - 4q20 - 2q22 - 2q26 + q28

HOMFLY-PT Polynomial:

2a-7z-1 + a-7z - 2a-7z3 - a-7z5 - 5a-5z-1 - 6a-5z + 3a-5z5 + a-5z7 + 3a-3z-1 + 6a-3z + 6a-3z3 + 4a-3z5 + a-3z7 - a-1z - 3a-1z3 - a-1z5

Kauffman Polynomial:

a-11z3 - a-11z5 - a-10 - 2a-10z2 + 7a-10z4 - 4a-10z6 - a-9z3 + 9a-9z5 - 6a-9z7 - 2a-8z2 + 2a-8z4 + 7a-8z6 - 6a-8z8 - 2a-7z-1 + a-7z + 9a-7z3 - 12a-7z5 + 9a-7z7 - 5a-7z9 + 5a-6 - 3a-6z2 - 9a-6z4 + 9a-6z6 - 2a-6z8 - 2a-6z10 - 5a-5z-1 + 7a-5z + 13a-5z3 - 37a-5z5 + 28a-5z7 - 9a-5z9 + 5a-4 - 2a-4z2 - 13a-4z4 + 10a-4z6 - 2a-4z10 - 3a-3z-1 + 8a-3z - 6a-3z3 - 5a-3z5 + 10a-3z7 - 4a-3z9 - 6a-2z4 + 11a-2z6 - 4a-2z8 + 2a-1z - 8a-1z3 + 10a-1z5 - 3a-1z7 - z2 + 3z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 20                       1 j = 18                     3   j = 16                   4 1   j = 14                 8 3     j = 12               7 4       j = 10             9 8         j = 8           8 7           j = 6         5 9             j = 4       5 8               j = 2     2 7                 j = 0   1 3                   j = -2   2                     j = -4 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, 17]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 17]]
Out[4]=	PD[X[6, 1, 7, 2], X[18, 7, 19, 8], X[4, 19, 1, 20], X[14, 6, 15, 5], > X[8, 4, 9, 3], X[16, 10, 17, 9], X[10, 16, 11, 15], X[20, 12, 21, 11], > X[22, 14, 5, 13], X[12, 22, 13, 21], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {4, -1, 2, -5, 6, -7, 8, -10, 9, -4, 7, -6, 11, -2, > 3, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 q - ------- + 5 Sqrt[q] - 10 q + 13 q - 17 q + 16 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 15 q + 12 q - 7 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 4 6 8 10 12 14 16 18 20 -q + q + 3 q - q + 6 q + 3 q + 3 q + 4 q - 3 q + q - 4 q - 22 26 28 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 17]][a, z]
Out[8]=	3 3 3 5 5 2 5 3 z 6 z 6 z z 2 z 6 z 3 z z 3 z ---- - ---- + ---- + -- - --- + --- - - - ---- + ---- - ---- - -- + ---- + 7 5 3 7 5 3 a 7 3 a 7 5 a z a z a z a a a a a a a 5 5 7 7 4 z z z z > ---- - -- + -- + -- 3 a 5 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 17]][a, z]
Out[9]=	2 -10 5 5 2 5 3 z 7 z 8 z 2 z 2 2 z -a + -- + -- - ---- - ---- - ---- + -- + --- + --- + --- - z - ---- - 6 4 7 5 3 7 5 3 a 10 a a a z a z a z a a a a 2 2 2 3 3 3 3 3 3 4 2 z 3 z 2 z z z 9 z 13 z 6 z 8 z 4 7 z > ---- - ---- - ---- + --- - -- + ---- + ----- - ---- - ---- + 3 z + ---- + 8 6 4 11 9 7 5 3 a 10 a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 2 z 9 z 13 z 6 z z 9 z 12 z 37 z 5 z 10 z > ---- - ---- - ----- - ---- - --- + ---- - ----- - ----- - ---- + ----- - 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 6 4 z 7 z 9 z 10 z 11 z 6 z 9 z 28 z 10 z > z - ---- + ---- + ---- + ----- + ----- - ---- + ---- + ----- + ----- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a 7 8 8 8 9 9 9 10 10 3 z 6 z 2 z 4 z 5 z 9 z 4 z 2 z 2 z > ---- - ---- - ---- - ---- - ---- - ---- - ---- - ----- - ----- a 8 6 2 7 5 3 6 4 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 2 3 2 q 4 6 6 2 7 q + 5 q + ----- + t + ----- + - + ---- + 8 q t + 5 q t + 9 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 14 5 > 8 q t + 7 q t + 9 q t + 8 q t + 7 q t + 4 q t + 8 q t + 14 6 16 6 16 7 18 7 20 8 > 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a17

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