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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X16,8,17,7 X22,18,5,17 X18,12,19,11 X12,22,13,21 X20,14,21,13 X14,20,15,19 X8,16,9,15 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 3q-5/2 - 8q-3/2 + 12q-1/2 - 17q1/2 + 18q3/2 - 19q5/2 + 16q7/2 - 11q9/2 + 7q11/2 - 3q13/2 + q15/2

A2 (sl(3)) Invariant:

q-12 - 2q-8 + 4q-6 + q-4 + 7 + 4q4 + q6 - q8 + q10 - 6q12 + q14 - 3q18 + 2q20 - q24

HOMFLY-PT Polynomial:

a-7z + a-5z-1 - 2a-5z3 - a-3z-1 - a-3z - a-3z3 + a-3z5 - 2a-1z-1 - 4a-1z - 2a-1z3 + a-1z5 + 2az-1 + az - 2az3 + a3z

Kauffman Polynomial:

- 2a-8z2 + 3a-8z4 - a-8z6 + a-7z - 5a-7z3 + 8a-7z5 - 3a-7z7 - 4a-6 + 11a-6z2 - 15a-6z4 + 14a-6z6 - 5a-6z8 + a-5z-1 + a-5z - 6a-5z3 + 4a-5z5 + 5a-5z7 - 4a-5z9 - 9a-4 + 36a-4z2 - 56a-4z4 + 42a-4z6 - 12a-4z8 - a-4z10 + a-3z-1 + a-3z - 7a-3z3 + a-3z5 + 11a-3z7 - 8a-3z9 - 4a-2 + 19a-2z2 - 38a-2z4 + 35a-2z6 - 13a-2z8 - a-2z10 - 2a-1z-1 + 10a-1z - 19a-1z3 + 17a-1z5 - 3a-1z7 - 4a-1z9 + 2 - 5z2 + 4z4 + 5z6 - 6z8 - 2az-1 + 8az - 11az3 + 11az5 - 6az7 - a2z2 + 4a2z4 - 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                   5 1   j = 10                 6 2     j = 8               10 5       j = 6             9 6         j = 4           9 10           j = 2         8 9             j = 0       5 10               j = -2     3 7                 j = -4     5                   j = -6 1 3                     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, 32]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 32]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], > X[18, 12, 19, 11], X[12, 22, 13, 21], X[20, 14, 21, 13], X[14, 20, 15, 19], > X[8, 16, 9, 15], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 5, -6, 7, -8, 9, -3, 4, -5, > 8, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 3 8 12 3/2 5/2 7/2 -q + ---- - ---- + ------- - 17 Sqrt[q] + 18 q - 19 q + 16 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 11 q + 7 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 4 -4 4 6 8 10 12 14 18 20 7 + q - -- + -- + q + 4 q + q - q + q - 6 q + q - 3 q + 2 q - 8 6 q q 24 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 32]][a, z]
Out[8]=	3 3 3 1 1 2 2 a z z 4 z 3 2 z z 2 z ---- - ---- - --- + --- + -- - -- - --- + a z + a z - ---- - -- - ---- - 5 3 a z z 7 3 a 5 3 a a z a z a a a a 5 5 3 z z > 2 a z + -- + -- 3 a a
In[9]:=	Kauffman[Link[11, Alternating, 32]][a, z]
Out[9]=	4 9 4 1 1 2 2 a z z z 10 z 2 - -- - -- - -- + ---- + ---- - --- - --- + -- + -- + -- + ---- + 8 a z - 6 4 2 5 3 a z z 7 5 3 a a a a a z a z a a a 2 2 2 2 3 3 3 3 2 2 z 11 z 36 z 19 z 2 2 5 z 6 z 7 z > a z - 5 z - ---- + ----- + ----- + ----- - a z - ---- - ---- - ---- - 8 6 4 2 7 5 3 a a a a a a a 3 4 4 4 4 19 z 3 3 3 4 3 z 15 z 56 z 38 z 2 4 > ----- - 11 a z + 2 a z + 4 z + ---- - ----- - ----- - ----- + 4 a z + a 8 6 4 2 a a a a 5 5 5 5 6 6 6 8 z 4 z z 17 z 5 3 5 6 z 14 z 42 z > ---- + ---- + -- + ----- + 11 a z - a z + 5 z - -- + ----- + ----- + 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 35 z 2 6 3 z 5 z 11 z 3 z 7 8 5 z > ----- - 3 a z - ---- + ---- + ----- - ---- - 6 a z - 6 z - ---- - 2 7 5 3 a 6 a a a a a 8 8 9 9 9 10 10 12 z 13 z 4 z 8 z 4 z z z > ----- - ----- - ---- - ---- - ---- - --- - --- 4 2 5 3 a 4 2 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 3 5 3 5 7 2 10 + 8 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 9 q t + 8 4 6 4 6 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 9 q t + 10 q t + 9 q t + 6 q t + 10 q t + 5 q t + 6 q t + 10 5 12 5 12 6 14 6 16 7 > 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a32

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