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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X18,10,19,9 X8493 X14,6,15,5 X20,16,21,15 X22,18,5,17 X16,22,17,21 X10,20,11,19 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

q-3/2 - 3q-1/2 + 6q1/2 - 13q3/2 + 17q5/2 - 22q7/2 + 22q9/2 - 20q11/2 + 16q13/2 - 10q15/2 + 5q17/2 - q19/2

A2 (sl(3)) Invariant:

- q-4 + q-2 - 1 + 5q4 - q6 + 7q8 + 2q10 + q12 + 4q14 - 4q16 + 3q18 - 4q20 - 2q22 + q24 - 3q26 + q28

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-11z5 - a-10 + 5a-10z4 - 5a-10z6 - 3a-9z3 + 14a-9z5 - 10a-9z7 - a-8z4 + 14a-8z6 - 11a-8z8 - 2a-7z-1 + 3a-7z - 6a-7z3 + 12a-7z5 + a-7z7 - 7a-7z9 + 5a-6 - 6a-6z2 - 12a-6z4 + 28a-6z6 - 13a-6z8 - 2a-6z10 - 5a-5z-1 + 13a-5z - 14a-5z3 - 2a-5z5 + 17a-5z7 - 11a-5z9 + 5a-4 - 7a-4z2 - 9a-4z4 + 17a-4z6 - 6a-4z8 - 2a-4z10 - 3a-3z-1 + 14a-3z - 20a-3z3 + 10a-3z5 + 3a-3z7 - 4a-3z9 - 3a-2z2 + 7a-2z6 - 4a-2z8 + 4a-1z - 9a-1z3 + 9a-1z5 - 3a-1z7 - 2z2 + 3z4 - z6

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a14

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