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DrawMorseLink

PD Presentation:

X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X14,7,15,8 X22,15,7,16 X20,18,21,17 X18,12,19,11 X12,20,13,19 X4,13,5,14 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 7q-15/2 + 12q-13/2 - 17q-11/2 + 19q-9/2 - 20q-7/2 + 17q-5/2 - 13q-3/2 + 8q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

q-28 - q-26 + 2q-24 + q-22 - 2q-20 + 4q-18 - 2q-16 + 3q-14 + 2q-12 - q-10 + 4q-8 - 4q-6 + 2q-4 - 1 + 2q2 - q4

HOMFLY-PT Polynomial:

2az3 + az5 - 2a3z3 - 3a3z5 - a3z7 - a5z-1 - 6a5z - 7a5z3 - 4a5z5 - a5z7 + a7z-1 + 3a7z + 3a7z3 + a7z5

Kauffman Polynomial:

2z4 - z6 - 5az3 + 10az5 - 4az7 + 3a2z2 - 14a2z4 + 19a2z6 - 7a2z8 + a3z3 - 5a3z5 + 12a3z7 - 6a3z9 + 8a4z2 - 28a4z4 + 30a4z6 - 8a4z8 - 2a4z10 + a5z-1 - 6a5z + 16a5z3 - 27a5z5 + 26a5z7 - 11a5z9 - a6 + 9a6z2 - 22a6z4 + 21a6z6 - 7a6z8 - 2a6z10 + a7z-1 - 4a7z + 5a7z3 - 4a7z5 + 5a7z7 - 5a7z9 + 2a8z2 - 5a8z4 + 8a8z6 - 6a8z8 + a9z - 3a9z3 + 7a9z5 - 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                   5 1   j = -2                 8 3     j = -4               10 6       j = -6             10 7         j = -8           9 10           j = -10         8 10             j = -12       4 9               j = -14     3 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, 211]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 211]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 5, 17, 6], > X[14, 7, 15, 8], X[22, 15, 7, 16], X[20, 18, 21, 17], X[18, 12, 19, 11], > X[12, 20, 13, 19], X[4, 13, 5, 14], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -10, 4, -11}, > {5, -1, 2, -3, 8, -9, 10, -5, 6, -4, 7, -8, 9, -7, 11, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 7 12 17 19 20 17 13 -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 8 3/2 > ------- - 4 Sqrt[q] + q Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-28 -26 2 -22 2 4 2 3 2 -10 4 4 -1 + q - q + --- + q - --- + --- - --- + --- + --- - q + -- - -- + 24 20 18 16 14 12 8 6 q q q q q q q q 2 2 4 > -- + 2 q - q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 211]][a, z]
Out[8]=	5 7 a a 5 7 3 3 3 5 3 7 3 5 -(--) + -- - 6 a z + 3 a z + 2 a z - 2 a z - 7 a z + 3 a z + a z - z z 3 5 5 5 7 5 3 7 5 7 > 3 a z - 4 a z + a z - a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 211]][a, z]
Out[9]=	5 7 6 a a 5 7 9 11 2 2 4 2 6 2 -a + -- + -- - 6 a z - 4 a z + a z - a z + 3 a z + 8 a z + 9 a z + z z 8 2 10 2 3 3 3 5 3 7 3 9 3 > 2 a z - 2 a z - 5 a z + a z + 16 a z + 5 a z - 3 a z + 11 3 4 2 4 4 4 6 4 8 4 10 4 > 2 a z + 2 z - 14 a z - 28 a z - 22 a z - 5 a z + 5 a z + 5 3 5 5 5 7 5 9 5 11 5 6 2 6 > 10 a z - 5 a z - 27 a z - 4 a z + 7 a z - a z - z + 19 a z + 4 6 6 6 8 6 10 6 7 3 7 5 7 > 30 a z + 21 a z + 8 a z - 3 a z - 4 a z + 12 a z + 26 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 5 a z - 5 a z - 7 a z - 8 a z - 7 a z - 6 a z - 6 a z - 5 9 7 9 4 10 6 10 > 11 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 8 1 2 1 5 3 8 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 9 10 10 7 10 > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 5 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 L11a211

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