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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 4q-1/2 + 6q1/2 - 9q3/2 + 9q5/2 - 10q7/2 + 9q9/2 - 7q11/2 + 4q13/2 - 2q15/2 + q17/2

A2 (sl(3)) Invariant:

q-6 + 2q-2 + q2 + 2q4 + 4q8 - q10 + 2q12 + q18 - 2q20 - q24

HOMFLY-PT Polynomial:

a-5z-1 + 8a-5z + 12a-5z3 + 6a-5z5 + a-5z7 - 3a-3z-1 - 17a-3z - 32a-3z3 - 24a-3z5 - 8a-3z7 - a-3z9 + 2a-1z-1 + 9a-1z + 12a-1z3 + 6a-1z5 + a-1z7

Kauffman Polynomial:

2a-10z2 - a-10z4 + 3a-9z3 - 2a-9z5 - 3a-8z2 + 5a-8z4 - 3a-8z6 + 2a-7z - 12a-7z3 + 10a-7z5 - 4a-7z7 - a-6 + 6a-6z2 - 17a-6z4 + 12a-6z6 - 4a-6z8 + a-5z-1 - 8a-5z + 17a-5z3 - 18a-5z5 + 10a-5z7 - 3a-5z9 - 3a-4 + 17a-4z2 - 22a-4z4 + 12a-4z6 - a-4z8 - a-4z10 + 3a-3z-1 - 21a-3z + 51a-3z3 - 46a-3z5 + 23a-3z7 - 5a-3z9 - 3a-2 + 9a-2z2 - 10a-2z4 + 6a-2z6 + a-2z8 - a-2z10 + 2a-1z-1 - 9a-1z + 12a-1z3 - 11a-1z5 + 8a-1z7 - 2a-1z9 + 3z2 - 11z4 + 9z6 - 2z8 + 2az - 7az3 + 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                   3 1   j = 12                 4 1     j = 10               5 3       j = 8             5 4         j = 6           4 5           j = 4         5 5             j = 2       3 6               j = 0     1 3                 j = -2   1 3                   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, 301]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 301]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[4, 9, 5, 10], > X[16, 6, 17, 5], X[20, 14, 21, 13], X[22, 16, 9, 15], X[14, 22, 15, 21], > X[18, 8, 19, 7], X[6, 18, 7, 17], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -4, 5, -10, 9, -11}, > {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 4 3/2 5/2 7/2 9/2 -q + ---- - ------- + 6 Sqrt[q] - 9 q + 9 q - 10 q + 9 q - 3/2 Sqrt[q] q 11/2 13/2 15/2 17/2 > 7 q + 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 4 8 10 12 18 20 24 q + -- + q + 2 q + 4 q - q + 2 q + q - 2 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 301]][a, z]
Out[8]=	3 3 3 5 5 1 3 2 8 z 17 z 9 z 12 z 32 z 12 z 6 z 24 z ---- - ---- + --- + --- - ---- + --- + ----- - ----- + ----- + ---- - ----- + 5 3 a z 5 3 a 5 3 a 5 3 a z 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, 301]][a, z]
Out[9]=	-6 3 3 1 3 2 2 z 8 z 21 z 9 z 2 -a - -- - -- + ---- + ---- + --- + --- - --- - ---- - --- + 2 a z + 3 z + 4 2 5 3 a z 7 5 3 a a a a z a z a a a 2 2 2 2 2 3 3 3 3 3 2 z 3 z 6 z 17 z 9 z 3 z 12 z 17 z 51 z 12 z > ---- - ---- + ---- + ----- + ---- + ---- - ----- + ----- + ----- + ----- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 4 4 4 4 4 5 5 3 4 z 5 z 17 z 22 z 10 z 2 z 10 z > 7 a z - 11 z - --- + ---- - ----- - ----- - ----- - ---- + ----- - 10 8 6 4 2 9 7 a a a a a a a 5 5 5 6 6 6 6 18 z 46 z 11 z 5 6 3 z 12 z 12 z 6 z > ----- - ----- - ----- + 5 a z + 9 z - ---- + ----- + ----- + ---- - 5 3 a 8 6 4 2 a a a a a a 7 7 7 7 8 8 8 9 9 4 z 10 z 23 z 8 z 7 8 4 z z z 3 z 5 z > ---- + ----- + ----- + ---- - a z - 2 z - ---- - -- + -- - ---- - ---- - 7 5 3 a 6 4 2 5 3 a a a a a a a a 9 10 10 2 z z z > ---- - --- - --- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 3 3 3 q 4 6 q + 5 q + ----- + ----- + ----- + t + ----- + - + ---- + 5 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 > 4 q t + 5 q t + 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + 12 5 14 5 14 6 16 6 18 7 > q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a301

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