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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-23/2 - 2q-21/2 + 3q-19/2 - 5q-17/2 + 6q-15/2 - 7q-13/2 + 6q-11/2 - 5q-9/2 + 4q-7/2 - 3q-5/2 + q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 + q-26 + 2q-22 + q-18 + q-16 - q-14 + q-12 + 2q-8 + q-6 + q-4 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 6a3z - 5a3z3 - a3z5 + a5z-1 + 4a5z + 7a5z3 + 5a5z5 + a5z7 + 3a7z + 7a7z3 + 5a7z5 + a7z7 - 3a9z - 4a9z3 - a9z5

Kauffman Polynomial:

- a3z-1 + 7a3z - 11a3z3 + 6a3z5 - a3z7 + a4 - 3a4z2 - 2a4z4 + 4a4z6 - a4z8 - a5z-1 + 5a5z - 8a5z3 + 2a5z5 + 3a5z7 - a5z9 - 8a6z2 + 16a6z4 - 11a6z6 + 5a6z8 - a6z10 - 5a7z + 22a7z3 - 27a7z5 + 15a7z7 - 3a7z9 - a8z2 + 6a8z4 - 7a8z6 + 4a8z8 - a8z10 - 2a9z + 11a9z3 - 17a9z5 + 9a9z7 - 2a9z9 + a10z2 - 8a10z4 + 6a10z6 - 2a10z8 - 4a11z3 + 4a11z5 - 2a11z7 - a12z2 + 3a12z4 - 2a12z6 - a13z + 4a13z3 - 2a13z5 + 2a14z2 - a14z4

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                       1 j = -2                         j = -4                   3 1   j = -6                 2 1     j = -8               3 2       j = -10             3 2         j = -12           4 3           j = -14         3 4             j = -16       2 3               j = -18     1 3                 j = -20   1 2                   j = -22   1                     j = -24 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, 278]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 278]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[8, 9, 1, 10], X[4, 13, 5, 14], X[20, 17, 21, 18], X[18, 8, 19, 7], > X[6, 20, 7, 19], X[22, 15, 9, 16], X[16, 21, 17, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 4, -9, 8, -5}, > {5, -1, 2, -3, 6, -4, 10, -11, 7, -8, 9, -7, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 2 3 5 6 7 6 5 4 3 q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 q q q q q q q q q -(3/2) 1 > q - ------- Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -26 2 -18 -16 -14 -12 2 -6 -4 -2 -q + q + --- + q + q - q + q + -- + q + q + q 22 8 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 278]][a, z]
Out[8]=	3 5 a a 3 5 7 9 3 3 5 3 7 3 -(--) + -- - 6 a z + 4 a z + 3 a z - 3 a z - 5 a z + 7 a z + 7 a z - z z 9 3 3 5 5 5 7 5 9 5 5 7 7 7 > 4 a z - a z + 5 a z + 5 a z - a z + a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 278]][a, z]
Out[9]=	3 5 4 a a 3 5 7 9 13 4 2 6 2 a - -- - -- + 7 a z + 5 a z - 5 a z - 2 a z - a z - 3 a z - 8 a z - z z 8 2 10 2 12 2 14 2 3 3 5 3 7 3 > a z + a z - a z + 2 a z - 11 a z - 8 a z + 22 a z + 9 3 11 3 13 3 4 4 6 4 8 4 10 4 > 11 a z - 4 a z + 4 a z - 2 a z + 16 a z + 6 a z - 8 a z + 12 4 14 4 3 5 5 5 7 5 9 5 11 5 > 3 a z - a z + 6 a z + 2 a z - 27 a z - 17 a z + 4 a z - 13 5 4 6 6 6 8 6 10 6 12 6 3 7 > 2 a z + 4 a z - 11 a z - 7 a z + 6 a z - 2 a z - a z + 5 7 7 7 9 7 11 7 4 8 6 8 8 8 > 3 a z + 15 a z + 9 a z - 2 a z - a z + 5 a z + 4 a z - 10 8 5 9 7 9 9 9 6 10 8 10 > 2 a z - a z - 3 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 3 1 1 1 2 1 3 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q q t q t q t q t q t q t q t 3 3 4 4 3 3 2 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 q t q t q t q t q t q t q t q t 2 2 t 2 > ---- + ---- + -- + t 8 6 4 q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a278

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