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DrawMorseLink

PD Presentation:

X8192 X20,9,21,10 X4758 X16,5,17,6 X6,15,1,16 X22,17,7,18 X18,13,19,14 X14,21,15,22 X2,11,3,12 X12,3,13,4 X10,19,11,20

Gauss Code:

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

Jones Polynomial:

q-27/2 - 4q-25/2 + 10q-23/2 - 17q-21/2 + 23q-19/2 - 27q-17/2 + 27q-15/2 - 24q-13/2 + 17q-11/2 - 11q-9/2 + 4q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 - q-40 + 2q-38 - 4q-36 + 4q-32 - 4q-30 + 5q-28 - q-26 + q-24 + 4q-22 - 3q-20 + 7q-18 - 2q-16 + 4q-12 - 3q-10 + q-8

HOMFLY-PT Polynomial:

- a5z3 - a5z5 - a7z-1 - 7a7z - 10a7z3 - 4a7z5 - a9z - 5a9z3 - 3a9z5 + 2a11z-1 + 6a11z + 4a11z3 - a13z-1 - a13z

Kauffman Polynomial:

a5z3 - a5z5 + 3a6z4 - 4a6z6 - a7z-1 + 7a7z - 14a7z3 + 16a7z5 - 10a7z7 + a8 - a8z2 - 9a8z4 + 19a8z6 - 13a8z8 - 6a9z3 + 9a9z5 + 6a9z7 - 10a9z9 - 3a10 + 17a10z2 - 46a10z4 + 56a10z6 - 21a10z8 - 3a10z10 + 2a11z-1 - 9a11z + 10a11z3 - 13a11z5 + 30a11z7 - 18a11z9 - 5a12 + 25a12z2 - 49a12z4 + 51a12z6 - 16a12z8 - 3a12z10 + a13z-1 - a13z - 4a13z3 + 3a13z5 + 10a13z7 - 8a13z9 - 2a14 + 6a14z2 - 13a14z4 + 17a14z6 - 8a14z8 + a15z - 5a15z3 + 8a15z5 - 4a15z7 - a16z2 + 2a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                       1 j = -6                     4 1 j = -8                   7     j = -10                 10 4     j = -12               14 7       j = -14             13 10         j = -16           14 14           j = -18         10 14             j = -20       7 13               j = -22     3 10                 j = -24   1 7                   j = -26   3                     j = -28 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, 226]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 226]]
Out[4]=	PD[X[8, 1, 9, 2], X[20, 9, 21, 10], X[4, 7, 5, 8], X[16, 5, 17, 6], > X[6, 15, 1, 16], X[22, 17, 7, 18], X[18, 13, 19, 14], X[14, 21, 15, 22], > X[2, 11, 3, 12], X[12, 3, 13, 4], X[10, 19, 11, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 10, -3, 4, -5}, > {3, -1, 2, -11, 9, -10, 7, -8, 5, -4, 6, -7, 11, -2, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 4 10 17 23 27 27 24 17 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 11 4 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 2 4 4 4 5 -26 -24 4 3 7 -q - q + --- - --- + --- - --- + --- - q + q + --- - --- + --- - 38 36 32 30 28 22 20 18 q q q q q q q q 2 4 3 -8 > --- + --- - --- + q 16 12 10 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 226]][a, z]
Out[8]=	7 11 13 a 2 a a 7 9 11 13 5 3 7 3 -(--) + ----- - --- - 7 a z - a z + 6 a z - a z - a z - 10 a z - z z z 9 3 11 3 5 5 7 5 9 5 > 5 a z + 4 a z - a z - 4 a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 226]][a, z]
Out[9]=	7 11 13 8 10 12 14 a 2 a a 7 11 13 a - 3 a - 5 a - 2 a - -- + ----- + --- + 7 a z - 9 a z - a z + z z z 15 8 2 10 2 12 2 14 2 16 2 5 3 > a z - a z + 17 a z + 25 a z + 6 a z - a z + a z - 7 3 9 3 11 3 13 3 15 3 6 4 8 4 > 14 a z - 6 a z + 10 a z - 4 a z - 5 a z + 3 a z - 9 a z - 10 4 12 4 14 4 16 4 5 5 7 5 9 5 > 46 a z - 49 a z - 13 a z + 2 a z - a z + 16 a z + 9 a z - 11 5 13 5 15 5 6 6 8 6 10 6 > 13 a z + 3 a z + 8 a z - 4 a z + 19 a z + 56 a z + 12 6 14 6 16 6 7 7 9 7 11 7 > 51 a z + 17 a z - a z - 10 a z + 6 a z + 30 a z + 13 7 15 7 8 8 10 8 12 8 14 8 > 10 a z - 4 a z - 13 a z - 21 a z - 16 a z - 8 a z - 9 9 11 9 13 9 10 10 12 10 > 10 a z - 18 a z - 8 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 3 1 7 3 10 7 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 13 10 14 14 14 13 10 14 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 7 10 4 7 4 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a226

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