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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3/2 - 3q-1/2 + 5q1/2 - 9q3/2 + 12q5/2 - 15q7/2 + 14q9/2 - 13q11/2 + 10q13/2 - 6q15/2 + 3q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + q-2 + 2q4 - 2q6 + 4q8 + q10 + 2q12 + 3q14 - 2q16 + 2q18 - 2q20 + q24 - q26 + q28

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-11z + 2a-11z3 - a-11z5 - 3a-10z2 + 6a-10z4 - 3a-10z6 - a-9z + a-9z3 + 5a-9z5 - 4a-9z7 + a-8z2 + 4a-8z6 - 4a-8z8 - 2a-7z + 7a-7z3 - 5a-7z5 + 3a-7z7 - 3a-7z9 + 8a-6z2 - 19a-6z4 + 15a-6z6 - 5a-6z8 - a-6z10 - a-5z-1 + a-5z + 6a-5z3 - 17a-5z5 + 15a-5z7 - 6a-5z9 + a-4 + 7a-4z2 - 25a-4z4 + 21a-4z6 - 5a-4z8 - a-4z10 - a-3z-1 + 5a-3z - 10a-3z3 + 4a-3z5 + 5a-3z7 - 3a-3z9 + 2a-2z2 - 9a-2z4 + 12a-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                     2   j = 16                   4 1   j = 14                 6 2     j = 12               7 4       j = 10             8 7         j = 8           7 6           j = 6         5 8             j = 4       4 7               j = 2     2 6                 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, 305]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 305]]
Out[4]=	PD[X[10, 1, 11, 2], X[22, 11, 9, 12], X[8, 9, 1, 10], X[2, 22, 3, 21], > X[12, 4, 13, 3], X[14, 20, 15, 19], X[18, 8, 19, 7], X[16, 6, 17, 5], > X[4, 16, 5, 15], X[6, 18, 7, 17], X[20, 14, 21, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 5, -9, 8, -10, 7, -3}, > {3, -1, 2, -5, 11, -6, 9, -8, 10, -7, 6, -11, 4, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 q - ------- + 5 Sqrt[q] - 9 q + 12 q - 15 q + 14 q - Sqrt[q] 11/2 13/2 15/2 17/2 19/2 > 13 q + 10 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 4 6 8 10 12 14 16 18 20 -q + q + 2 q - 2 q + 4 q + q + 2 q + 3 q - 2 q + 2 q - 2 q + 24 26 28 > q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 305]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 2 z 2 z 3 z z 3 z 5 z 5 z 3 z z 4 z -(----) + ---- - --- + --- + --- - - - ---- + ---- + ---- - ---- - -- + ---- + 5 3 7 5 3 a 7 5 3 a 7 5 a z a z a 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, 305]][a, z]
Out[9]=	2 2 2 -4 1 1 z z 2 z z 5 z 2 z 2 3 z z 8 z a - ---- - ---- - --- - -- - --- + -- + --- + --- - z - ---- + -- + ---- + 5 3 11 9 7 5 3 a 10 8 6 a z a z a a a a a a a a 2 2 3 3 3 3 3 3 4 7 z 2 z 2 z z 7 z 6 z 10 z 8 z 4 6 z > ---- + ---- + ---- + -- + ---- + ---- - ----- - ---- + 3 z + ---- - 4 2 11 9 7 5 3 a 10 a a a a a a a a 4 4 4 5 5 5 5 5 5 19 z 25 z 9 z z 5 z 5 z 17 z 4 z 10 z 6 > ----- - ----- - ---- - --- + ---- - ---- - ----- + ---- + ----- - z - 6 4 2 11 9 7 5 3 a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 3 z 4 z 15 z 21 z 12 z 4 z 3 z 15 z 5 z 3 z > ---- + ---- + ----- + ----- + ----- - ---- + ---- + ----- + ---- - ---- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 4 z 5 z 5 z 4 z 3 z 6 z 3 z z z > ---- - ---- - ---- - ---- - ---- - ---- - ---- - --- - --- 8 6 4 2 7 5 3 6 4 a 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 6 q + 4 q + ----- + t + ----- + - + ---- + 7 q t + 5 q t + 8 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 > 7 q t + 6 q t + 8 q t + 7 q t + 7 q t + 4 q t + 6 q t + 14 6 16 6 16 7 18 7 20 8 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a305

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