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L11n193 L11n195

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DrawMorseLink

PD Presentation:

X8192 X16,6,17,5 X9,20,10,21 X21,10,22,11 X18,16,19,15 X14,20,15,19 X2,11,3,12 X12,3,13,4 X4758 X22,14,7,13 X6,18,1,17

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 3q-11/2 - 6q-9/2 + 9q-7/2 - 11q-5/2 + 11q-3/2 - 11q-1/2 + 7q1/2 - 5q3/2 + 2q5/2

A2 (sl(3)) Invariant:

q-20 - q-18 + 2q-14 - 2q-12 + 2q-10 + 3q-4 + 5 + q6 - 2q8

HOMFLY-PT Polynomial:

3a-1z + 2a-1z3 - az-1 - 6az - 6az3 - 2az5 + a3z-1 + a3z - a3z3 - a3z5 + a5z + a5z3

Kauffman Polynomial:

4a-2z2 - 3a-2z4 - 5a-1z + 8a-1z3 - 3a-1z5 - a-1z7 + 5z2 - 5z4 + 2z6 - 2z8 + az-1 - 9az + 11az3 - 2az5 - 2az7 - az9 - a2 + 3a2z2 - 4a2z4 + 6a2z6 - 5a2z8 + a3z-1 - 3a3z + 8a3z5 - 5a3z7 - a3z9 - a4z2 + 4a4z4 + a4z6 - 3a4z8 - a5z3 + 6a5z5 - 4a5z7 - 3a6z2 + 6a6z4 - 3a6z6 - a7z + 2a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 6                   2 j = 4                 3   j = 2               4 2   j = 0             7 3     j = -2           5 5       j = -4         6 6         j = -6       4 6           j = -8     2 5             j = -10   1 4               j = -12   2                 j = -14 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, NonAlternating, 194]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 194]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 6, 17, 5], X[9, 20, 10, 21], X[21, 10, 22, 11], > X[18, 16, 19, 15], X[14, 20, 15, 19], X[2, 11, 3, 12], X[12, 3, 13, 4], > X[4, 7, 5, 8], X[22, 14, 7, 13], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -7, 8, -9, 2, -11}, > {9, -1, -3, 4, 7, -8, 10, -6, 5, -2, 11, -5, 6, 3, -4, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 6 9 11 11 11 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 7 Sqrt[q] - 5 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 > 2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 2 2 2 3 6 8 5 + q - q + --- - --- + --- + -- + q - 2 q 14 12 10 4 q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 194]][a, z]
Out[8]=	3 3 a a 3 z 3 5 2 z 3 3 3 5 3 -(-) + -- + --- - 6 a z + a z + a z + ---- - 6 a z - a z + a z - z z a a 5 3 5 > 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 194]][a, z]
Out[9]=	3 2 2 a a 5 z 3 7 2 4 z 2 2 4 2 -a + - + -- - --- - 9 a z - 3 a z - a z + 5 z + ---- + 3 a z - a z - z z a 2 a 3 4 6 2 8 z 3 5 3 7 3 4 3 z 2 4 > 3 a z + ---- + 11 a z - a z + 2 a z - 5 z - ---- - 4 a z + a 2 a 5 4 4 6 4 3 z 5 3 5 5 5 7 5 6 > 4 a z + 6 a z - ---- - 2 a z + 8 a z + 6 a z - a z + 2 z + a 7 2 6 4 6 6 6 z 7 3 7 5 7 8 > 6 a z + a z - 3 a z - -- - 2 a z - 5 a z - 4 a z - 2 z - a 2 8 4 8 9 3 9 > 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 2 1 4 2 5 4 6 7 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 6 6 5 2 2 2 4 2 6 3 > ----- + ---- + ---- + 3 t + 4 q t + 2 q t + 3 q t + 2 q t 4 2 4 2 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n194

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