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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 3q-1/2 + 4q1/2 - 6q3/2 + 6q5/2 - 7q7/2 + 6q9/2 - 5q11/2 + 3q13/2 - 2q15/2 + q17/2

A2 (sl(3)) Invariant:

q-6 + q-2 + q4 + 3q8 + 2q12 + q14 + q16 + q18 - q20 - q24

HOMFLY-PT Polynomial:

6a-5z + 11a-5z3 + 6a-5z5 + a-5z7 - a-3z-1 - 12a-3z - 28a-3z3 - 23a-3z5 - 8a-3z7 - a-3z9 + a-1z-1 + 7a-1z + 11a-1z3 + 6a-1z5 + a-1z7

Kauffman Polynomial:

2a-10z2 - a-10z4 - a-9z + 4a-9z3 - 2a-9z5 - a-8z2 + 3a-8z4 - 2a-8z6 - 4a-7z3 + 4a-7z5 - 2a-7z7 + a-6z2 - 8a-6z4 + 6a-6z6 - 2a-6z8 - 5a-5z + 15a-5z3 - 18a-5z5 + 9a-5z7 - 2a-5z9 + 5a-4z2 - 5a-4z4 - a-4z6 + 3a-4z8 - a-4z10 + a-3z-1 - 14a-3z + 43a-3z3 - 45a-3z5 + 22a-3z7 - 4a-3z9 - a-2 + 5a-2z2 - 6a-2z4 + a-2z6 + 3a-2z8 - a-2z10 + a-1z-1 - 7a-1z + 14a-1z3 - 16a-1z5 + 10a-1z7 - 2a-1z9 + 4z2 - 13z4 + 10z6 - 2z8 + az - 6az3 + 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 r = 7 j = 18                       1 j = 16                     1   j = 14                   2 1   j = 12                 3 1     j = 10               3 2       j = 8             4 3         j = 6           3 4           j = 4         3 3             j = 2       2 4               j = 0     1 2                 j = -2   1 2                   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[11, Alternating, 364]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 364]]
Out[4]=	PD[X[12, 1, 13, 2], X[2, 13, 3, 14], X[14, 3, 15, 4], X[4, 11, 5, 12], > X[16, 6, 17, 5], X[22, 16, 11, 15], X[18, 8, 19, 7], X[20, 10, 21, 9], > X[6, 18, 7, 17], X[8, 20, 9, 19], X[10, 22, 1, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -9, 7, -10, 8, -11}, > {4, -1, 2, -3, 6, -5, 9, -7, 10, -8, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 3 3/2 5/2 7/2 9/2 -q + ---- - ------- + 4 Sqrt[q] - 6 q + 6 q - 7 q + 6 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 17/2 > 5 q + 3 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 -2 4 8 12 14 16 18 20 24 q + q + q + 3 q + 2 q + q + q + q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 364]][a, z]
Out[8]=	3 3 3 5 5 1 1 6 z 12 z 7 z 11 z 28 z 11 z 6 z 23 z -(----) + --- + --- - ---- + --- + ----- - ----- + ----- + ---- - ----- + 3 a z 5 3 a 5 3 a 5 3 a z a a a a a a 5 7 7 7 9 6 z z 8 z z z > ---- + -- - ---- + -- - -- a 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 364]][a, z]
Out[9]=	2 2 2 -2 1 1 z 5 z 14 z 7 z 2 2 z z z -a + ---- + --- - -- - --- - ---- - --- + a z + 4 z + ---- - -- + -- + 3 a z 9 5 3 a 10 8 6 a z a a a a a a 2 2 3 3 3 3 3 4 5 z 5 z 4 z 4 z 15 z 43 z 14 z 3 4 z > ---- + ---- + ---- - ---- + ----- + ----- + ----- - 6 a z - 13 z - --- + 4 2 9 7 5 3 a 10 a a a a a a a 4 4 4 4 5 5 5 5 5 3 z 8 z 5 z 6 z 2 z 4 z 18 z 45 z 16 z 5 > ---- - ---- - ---- - ---- - ---- + ---- - ----- - ----- - ----- + 5 a z + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 7 6 2 z 6 z z z 2 z 9 z 22 z 10 z 7 8 > 10 z - ---- + ---- - -- + -- - ---- + ---- + ----- + ----- - a z - 2 z - 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 2 z 3 z 3 z 2 z 4 z 2 z z 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 2 4 1 1 1 -2 2 2 2 q 4 4 q + 3 q + ----- + ----- + ----- + t + ----- + - + ---- + 3 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 > 3 q t + 4 q t + 4 q t + 3 q t + 3 q t + 2 q t + 3 q t + 12 5 14 5 14 6 16 6 18 7 > q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a364

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