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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 7q-15/2 + 12q-13/2 - 16q-11/2 + 18q-9/2 - 19q-7/2 + 16q-5/2 - 12q-3/2 + 7q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-30 + q-28 - q-26 + 2q-24 - 3q-20 + 2q-18 - 3q-16 + 2q-14 + 2q-12 + 5q-8 - 3q-6 + 3q-4 + q-2 - 1 + 2q2 - q4

HOMFLY-PT Polynomial:

2az3 + az5 - 2a3z-1 - 6a3z - 7a3z3 - 4a3z5 - a3z7 + 4a5z-1 + 9a5z + 9a5z3 + 3a5z5 - 3a7z-1 - 6a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial:

2z4 - z6 - 7az3 + 11az5 - 4az7 + 2a2z2 - 12a2z4 + 16a2z6 - 6a2z8 - 2a3z-1 + 8a3z - 17a3z3 + 14a3z5 + 2a3z7 - 4a3z9 + 2a4 + 2a4z2 - 22a4z4 + 30a4z6 - 11a4z8 - a4z10 - 4a5z-1 + 17a5z - 26a5z3 + 17a5z5 + 4a5z7 - 7a5z9 + 3a6 - 5a6z2 - 6a6z4 + 18a6z6 - 10a6z8 - a6z10 - 3a7z-1 + 12a7z - 22a7z3 + 22a7z5 - 7a7z7 - 3a7z9 + 3a8 - 8a8z2 + 7a8z4 + 2a8z6 - 5a8z8 - a9z-1 + 2a9z - 4a9z3 + 7a9z5 - 5a9z7 + a10 - 3a10z2 + 5a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

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 = 4                       1 j = 2                     3   j = 0                   4 1   j = -2                 8 3     j = -4               9 5       j = -6             10 7         j = -8           8 9           j = -10         8 10             j = -12       5 9               j = -14     2 7                 j = -16   1 5                   j = -18   2                     j = -20 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, 106]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 106]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[16, 8, 17, 7], X[22, 18, 5, 17], > X[18, 9, 19, 10], X[8, 21, 9, 22], X[20, 11, 21, 12], X[10, 19, 11, 20], > X[12, 16, 13, 15], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -6, 5, -8, 7, -9, 11, -2, 9, -3, 4, -5, > 8, -7, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 7 12 16 18 19 16 12 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 7 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 2 3 2 3 2 2 5 3 3 -1 + q + q - q + --- - --- + --- - --- + --- + --- + -- - -- + -- + 24 20 18 16 14 12 8 6 4 q q q q q q q q q -2 2 4 > q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 106]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 9 3 3 3 ----- + ---- - ---- + -- - 6 a z + 9 a z - 6 a z + a z + 2 a z - 7 a z + z z z z 5 3 7 3 5 3 5 5 5 3 7 > 9 a z - 3 a z + a z - 4 a z + 3 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 106]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 4 a 3 a a 3 5 2 a + 3 a + 3 a + a - ---- - ---- - ---- - -- + 8 a z + 17 a z + z z z z 7 9 11 2 2 4 2 6 2 8 2 > 12 a z + 2 a z - a z + 2 a z + 2 a z - 5 a z - 8 a z - 10 2 3 3 3 5 3 7 3 9 3 11 3 > 3 a z - 7 a z - 17 a z - 26 a z - 22 a z - 4 a z + 2 a z + 4 2 4 4 4 6 4 8 4 10 4 5 > 2 z - 12 a z - 22 a z - 6 a z + 7 a z + 5 a z + 11 a z + 3 5 5 5 7 5 9 5 11 5 6 2 6 > 14 a z + 17 a z + 22 a z + 7 a z - a z - z + 16 a z + 4 6 6 6 8 6 10 6 7 3 7 5 7 > 30 a z + 18 a z + 2 a z - 3 a z - 4 a z + 2 a z + 4 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 7 a z - 5 a z - 6 a z - 11 a z - 10 a z - 5 a z - 4 a z - 5 9 7 9 4 10 6 10 > 7 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 8 1 2 1 5 2 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 9 8 10 8 9 10 7 9 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 4 t + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 q t q t q t q t q t q t q t q t 3 t 2 2 2 4 3 > --- + t + 3 q t + q t 2 q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a106

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