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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 7q-15/2 + 13q-13/2 - 18q-11/2 + 21q-9/2 - 22q-7/2 + 18q-5/2 - 15q-3/2 + 9q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

az + 2az3 + az5 - a3z-1 - 2a3z - 3a3z3 - 3a3z5 - a3z7 + a5z-1 - 4a5z - 7a5z3 - 4a5z5 - a5z7 + 3a7z + 3a7z3 + a7z5

Kauffman Polynomial:

- z2 + 2z4 - z6 + 2az - 6az3 + 9az5 - 4az7 + 2a2z2 - 7a2z4 + 15a2z6 - 7a2z8 - a3z-1 + 4a3z - 11a3z3 + 11a3z5 + 6a3z7 - 6a3z9 + a4 + 9a4z2 - 33a4z4 + 39a4z6 - 12a4z8 - 2a4z10 - a5z-1 - 2a5z3 - 9a5z5 + 22a5z7 - 12a5z9 + 15a6z2 - 42a6z4 + 38a6z6 - 12a6z8 - 2a6z10 - a7z - 4a7z5 + 7a7z7 - 6a7z9 + 7a8z2 - 13a8z4 + 12a8z6 - 7a8z8 - a9z3 + 6a9z5 - 5a9z7 - 2a10z2 + 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                   6 1   j = -2                 9 3     j = -4               10 7       j = -6             12 8         j = -8           9 10           j = -10         9 12             j = -12       5 10               j = -14     2 8                 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, 338]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 338]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[14, 3, 15, 4], X[20, 13, 21, 14], > X[12, 21, 13, 22], X[22, 5, 9, 6], X[16, 8, 17, 7], X[18, 16, 19, 15], > X[8, 18, 1, 17], X[6, 9, 7, 10], X[4, 19, 5, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 6, -10, 7, -9}, > {10, -1, 2, -5, 4, -3, 8, -7, 9, -8, 11, -4, 5, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 7 13 18 21 22 18 15 -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 9 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 4 3 3 3 3 6 3 3 2 4 -2 + q - q + --- - --- + --- - --- + --- + --- + -- - -- + -- + 2 q - q 24 20 18 16 14 12 8 6 4 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 338]][a, z]
Out[8]=	3 5 a a 3 5 7 3 3 3 5 3 -(--) + -- + a z - 2 a z - 4 a z + 3 a z + 2 a z - 3 a z - 7 a z + z z 7 3 5 3 5 5 5 7 5 3 7 5 7 > 3 a z + a z - 3 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 338]][a, z]
Out[9]=	3 5 4 a a 3 7 11 2 2 2 4 2 a - -- - -- + 2 a z + 4 a z - a z - a z - z + 2 a z + 9 a z + z z 6 2 8 2 10 2 3 3 3 5 3 9 3 > 15 a z + 7 a z - 2 a z - 6 a z - 11 a z - 2 a z - a z + 11 3 4 2 4 4 4 6 4 8 4 10 4 > 2 a z + 2 z - 7 a z - 33 a z - 42 a z - 13 a z + 5 a z + 5 3 5 5 5 7 5 9 5 11 5 6 2 6 > 9 a z + 11 a z - 9 a z - 4 a z + 6 a z - a z - z + 15 a z + 4 6 6 6 8 6 10 6 7 3 7 5 7 > 39 a z + 38 a z + 12 a z - 3 a z - 4 a z + 6 a z + 22 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 7 a z - 5 a z - 7 a z - 12 a z - 12 a z - 7 a z - 6 a z - 5 9 7 9 4 10 6 10 > 12 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 2 1 5 2 8 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 10 9 12 9 10 12 8 10 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 6 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 L11a338

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