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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X20,10,21,9 X22,13,7,14 X14,21,15,22 X10,16,11,15 X18,5,19,6 X16,20,17,19 X2738 X4,11,5,12 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-17/2 + 3q-15/2 - 8q-13/2 + 12q-11/2 - 17q-9/2 + 20q-7/2 - 20q-5/2 + 18q-3/2 - 14q-1/2 + 8q1/2 - 4q3/2 + q5/2

A2 (sl(3)) Invariant:

q-28 + 2q-26 - q-24 + q-22 + 4q-20 - 3q-18 + 2q-16 + q-14 - 3q-12 + 3q-10 - 2q-8 + 3q-6 - 2q-2 + 5 - 3q2 + 2q6 - q8

HOMFLY-PT Polynomial:

a-1z3 - az - az5 - a3z-1 - 4a3z - 3a3z3 - 2a3z5 + 2a5z-1 + 6a5z + 5a5z3 - 2a7z-1 - 4a7z + a9z-1

Kauffman Polynomial:

- a-2z4 + 2a-1z3 - 4a-1z5 + 6z4 - 8z6 - az - 6az3 + 16az5 - 12az7 + 5a2z2 - 17a2z4 + 24a2z6 - 13a2z8 - a3z-1 + 5a3z - 17a3z3 + 16a3z5 + 6a3z7 - 8a3z9 + 17a4z2 - 53a4z4 + 57a4z6 - 16a4z8 - 2a4z10 - 2a5z-1 + 9a5z - 15a5z3 - 6a5z5 + 28a5z7 - 12a5z9 - a6 + 15a6z2 - 39a6z4 + 35a6z6 - 6a6z8 - 2a6z10 - 2a7z-1 + 7a7z - 12a7z3 + 2a7z5 + 9a7z7 - 4a7z9 + 3a8z2 - 10a8z4 + 10a8z6 - 3a8z8 - a9z-1 + 4a9z - 6a9z3 + 4a9z5 - a9z7

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                       1 j = 4                     3   j = 2                   5 1   j = 0                 9 3     j = -2               10 6       j = -4             10 8         j = -6           10 10           j = -8         7 10             j = -10       5 10               j = -12     3 7                 j = -14   1 6                   j = -16   2                     j = -18 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, 239]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 239]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 10, 21, 9], X[22, 13, 7, 14], > X[14, 21, 15, 22], X[10, 16, 11, 15], X[18, 5, 19, 6], X[16, 20, 17, 19], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 7, -11}, > {9, -1, 3, -6, 10, -2, 4, -5, 6, -8, 11, -7, 8, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 8 12 17 20 20 18 14 -q + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 8 Sqrt[q] - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 2 -24 -22 4 3 2 -14 3 3 2 3 5 + q + --- - q + q + --- - --- + --- + q - --- + --- - -- + -- - 26 20 18 16 12 10 8 6 q q q q q q q q 2 2 6 8 > -- - 3 q + 2 q - q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 239]][a, z]
Out[8]=	3 5 7 9 3 a 2 a 2 a a 3 5 7 z 3 3 -(--) + ---- - ---- + -- - a z - 4 a z + 6 a z - 4 a z + -- - 3 a z + z z z z a 5 3 5 3 5 > 5 a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 239]][a, z]
Out[9]=	3 5 7 9 6 a 2 a 2 a a 3 5 7 9 -a - -- - ---- - ---- - -- - a z + 5 a z + 9 a z + 7 a z + 4 a z + z z z z 3 2 2 4 2 6 2 8 2 2 z 3 3 3 > 5 a z + 17 a z + 15 a z + 3 a z + ---- - 6 a z - 17 a z - a 4 5 3 7 3 9 3 4 z 2 4 4 4 > 15 a z - 12 a z - 6 a z + 6 z - -- - 17 a z - 53 a z - 2 a 5 6 4 8 4 4 z 5 3 5 5 5 7 5 > 39 a z - 10 a z - ---- + 16 a z + 16 a z - 6 a z + 2 a z + a 9 5 6 2 6 4 6 6 6 8 6 7 > 4 a z - 8 z + 24 a z + 57 a z + 35 a z + 10 a z - 12 a z + 3 7 5 7 7 7 9 7 2 8 4 8 6 8 > 6 a z + 28 a z + 9 a z - a z - 13 a z - 16 a z - 6 a z - 8 8 3 9 5 9 7 9 4 10 6 10 > 3 a z - 8 a z - 12 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 2 1 6 3 7 5 9 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 2 18 8 16 7 14 7 14 6 12 6 12 5 10 5 q q t q t q t q t q t q t q t 10 7 10 10 10 10 8 10 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 3 t + 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q t q t q t q t q t q t q t q t 2 2 2 4 2 6 3 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a239

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