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L11a346 L11a348

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DrawMorseLink

PD Presentation:

X12,1,13,2 X8493 X20,14,21,13 X22,16,11,15 X14,22,15,21 X6,18,7,17 X18,8,19,7 X16,20,17,19 X10,6,1,5 X4,10,5,9 X2,11,3,12

Gauss Code:

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

Jones Polynomial:

- q1/2 + q3/2 - 5q5/2 + 7q7/2 - 12q9/2 + 15q11/2 - 15q13/2 + 15q15/2 - 12q17/2 + 8q19/2 - 4q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + q4 + 3q6 + 5q8 + 2q10 + 6q12 - q14 - q16 - 2q18 - 6q20 + q22 - 3q24 + 2q26 + q28 - q30 + 2q32 - q34

HOMFLY-PT Polynomial:

- a-9z-1 + 2a-9z3 + a-9z5 + 5a-7z-1 + 6a-7z - 3a-7z5 - a-7z7 - 8a-5z-1 - 14a-5z - 11a-5z3 - 5a-5z5 - a-5z7 + 4a-3z-1 + 8a-3z + 5a-3z3 + a-3z5

Kauffman Polynomial:

- a-14z4 + 2a-13z3 - 4a-13z5 - 2a-12z2 + 8a-12z4 - 8a-12z6 - a-11z - 4a-11z3 + 13a-11z5 - 10a-11z7 + 2a-10 - 3a-10z2 + 9a-10z6 - 8a-10z8 - a-9z-1 - 3a-9z + 5a-9z3 + 5a-9z5 - 4a-9z9 + 9a-8 - 20a-8z2 + 11a-8z4 + 10a-8z6 - 6a-8z8 - a-8z10 - 5a-7z-1 + 7a-7z - 3a-7z3 - 3a-7z5 + 10a-7z7 - 5a-7z9 + 14a-6 - 33a-6z2 + 25a-6z4 - 5a-6z6 + a-6z8 - a-6z10 - 8a-5z-1 + 21a-5z - 27a-5z3 + 15a-5z5 - a-5z7 - a-5z9 + 8a-4 - 14a-4z2 + 5a-4z4 + 2a-4z6 - a-4z8 - 4a-3z-1 + 12a-3z - 13a-3z3 + 6a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 24                       1 j = 22                     3   j = 20                   5 1   j = 18                 7 3     j = 16               8 5       j = 14             7 7         j = 12           8 8           j = 10         4 7             j = 8       3 8               j = 6     2 4                 j = 4   1 5                   j = 2                         j = 0 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, 347]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 347]]
Out[4]=	PD[X[12, 1, 13, 2], X[8, 4, 9, 3], X[20, 14, 21, 13], X[22, 16, 11, 15], > X[14, 22, 15, 21], X[6, 18, 7, 17], X[18, 8, 19, 7], X[16, 20, 17, 19], > X[10, 6, 1, 5], X[4, 10, 5, 9], X[2, 11, 3, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 9, -6, 7, -2, 10, -9}, > {11, -1, 3, -5, 4, -8, 6, -7, 8, -3, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 15/2 -Sqrt[q] + q - 5 q + 7 q - 12 q + 15 q - 15 q + 15 q - 17/2 19/2 21/2 23/2 > 12 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 4 6 8 10 12 14 16 18 20 22 q + q + 3 q + 5 q + 2 q + 6 q - q - q - 2 q - 6 q + q - 24 26 28 30 32 34 > 3 q + 2 q + q - q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 347]][a, z]
Out[8]=	3 3 3 5 1 5 8 4 6 z 14 z 8 z 2 z 11 z 5 z z -(----) + ---- - ---- + ---- + --- - ---- + --- + ---- - ----- + ---- + -- - 9 7 5 3 7 5 3 9 5 3 9 a z a z a z a z a a a a a a a 5 5 5 7 7 3 z 5 z z z z > ---- - ---- + -- - -- - -- 7 5 3 7 5 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 347]][a, z]
Out[9]=	2 9 14 8 1 5 8 4 z 3 z 7 z 21 z --- + -- + -- + -- - ---- - ---- - ---- - ---- - --- - --- + --- + ---- + 10 8 6 4 9 7 5 3 11 9 7 5 a a a a a z a z a z a z a a a a 2 2 2 2 2 3 3 3 3 12 z 2 z 3 z 20 z 33 z 14 z 2 z 4 z 5 z 3 z > ---- - ---- - ---- - ----- - ----- - ----- + ---- - ---- + ---- - ---- - 3 12 10 8 6 4 13 11 9 7 a a a a a a a a a a 3 3 4 4 4 4 4 5 5 5 27 z 13 z z 8 z 11 z 25 z 5 z 4 z 13 z 5 z > ----- - ----- - --- + ---- + ----- + ----- + ---- - ---- + ----- + ---- - 5 3 14 12 8 6 4 13 11 9 a a a a a a a a a a 5 5 5 6 6 6 6 6 7 7 3 z 15 z 6 z 8 z 9 z 10 z 5 z 2 z 10 z 10 z > ---- + ----- + ---- - ---- + ---- + ----- - ---- + ---- - ----- + ----- - 7 5 3 12 10 8 6 4 11 7 a a a a a a a a a a 7 7 8 8 8 8 9 9 9 10 10 z z 8 z 6 z z z 4 z 5 z z z z > -- - -- - ---- - ---- + -- - -- - ---- - ---- - -- - --- - --- 5 3 10 8 6 4 9 7 5 8 6 a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 4 6 -2 q 6 8 8 2 10 2 10 3 5 q + 2 q + t + -- + 4 q t + 3 q t + 8 q t + 4 q t + 7 q t + t 12 3 12 4 14 4 14 5 16 5 16 6 > 8 q t + 8 q t + 7 q t + 7 q t + 8 q t + 5 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 7 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a347

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