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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 4q-5/2 - 10q-3/2 + 16q-1/2 - 22q1/2 + 25q3/2 - 25q5/2 + 21q7/2 - 16q9/2 + 9q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - 2q-8 + 3q-6 + q-4 - 2q-2 + 6 - 4q2 + 2q4 - q6 - 2q8 + 5q10 - 3q12 + 5q14 + q16 - 2q18 + 2q20 - q22

HOMFLY-PT Polynomial:

- a-5z-1 + a-5z + 2a-5z3 + a-5z5 + 2a-3z-1 + a-3z - 3a-3z3 - 3a-3z5 - a-3z7 - 2a-1z-1 - 3a-1z - 4a-1z3 - 3a-1z5 - a-1z7 + az-1 + 2az + 2az3 + az5

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + a-7z - 6a-7z3 + 9a-7z5 - 4a-7z7 + a-6z2 - 7a-6z4 + 14a-6z6 - 7a-6z8 - a-5z-1 + 3a-5z - 5a-5z3 + 3a-5z5 + 9a-5z7 - 7a-5z9 + 6a-4z2 - 23a-4z4 + 32a-4z6 - 10a-4z8 - 3a-4z10 - 2a-3z-1 + 7a-3z - 5a-3z3 - 12a-3z5 + 29a-3z7 - 16a-3z9 - a-2 + 9a-2z2 - 34a-2z4 + 42a-2z6 - 15a-2z8 - 3a-2z10 - 2a-1z-1 + 9a-1z - 16a-1z3 + 9a-1z5 + 7a-1z7 - 9a-1z9 + 5z2 - 16z4 + 21z6 - 12z8 - az-1 + 4az - 9az3 + 14az5 - 9az7 + 4a2z4 - 4a2z6 + a3z3 - a3z5

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 16                       1 j = 14                     3   j = 12                   6 1   j = 10                 10 3     j = 8               11 6       j = 6             14 10         j = 4           12 12           j = 2         10 13             j = 0       7 13               j = -2     3 9                 j = -4   1 7                   j = -6   3                     j = -8 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, 178]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 178]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[16, 7, 17, 8], X[20, 10, 21, 9], > X[22, 15, 7, 16], X[14, 6, 15, 5], X[18, 14, 19, 13], X[12, 20, 13, 19], > X[4, 12, 5, 11], X[6, 18, 1, 17], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -9, 6, -10}, > {3, -1, 4, -2, 9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 10 16 3/2 5/2 7/2 -q + ---- - ---- + ------- - 22 Sqrt[q] + 25 q - 25 q + 21 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 16 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 3 -4 2 2 4 6 8 10 12 6 + q - -- + -- + q - -- - 4 q + 2 q - q - 2 q + 5 q - 3 q + 8 6 2 q q q 14 16 18 20 22 > 5 q + q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 178]][a, z]
Out[8]=	3 3 3 1 2 2 a z z 3 z 2 z 3 z 4 z -(----) + ---- - --- + - + -- + -- - --- + 2 a z + ---- - ---- - ---- + 5 3 a z z 5 3 a 5 3 a a z a z a a a a 5 5 5 7 7 3 z 3 z 3 z 5 z z > 2 a z + -- - ---- - ---- + a z - -- - -- 5 3 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 178]][a, z]
Out[9]=	2 2 -2 1 2 2 a z 3 z 7 z 9 z 2 z z -a - ---- - ---- - --- - - + -- + --- + --- + --- + 4 a z + 5 z - -- + -- + 5 3 a z z 7 5 3 a 8 6 a z a z a a a a a 2 2 3 3 3 3 4 6 z 9 z 6 z 5 z 5 z 16 z 3 3 3 4 2 z > ---- + ---- - ---- - ---- - ---- - ----- - 9 a z + a z - 16 z + ---- - 4 2 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 7 z 23 z 34 z 2 4 9 z 3 z 12 z 9 z 5 > ---- - ----- - ----- + 4 a z + ---- + ---- - ----- + ---- + 14 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 6 7 7 3 5 6 z 14 z 32 z 42 z 2 6 4 z 9 z > a z + 21 z - -- + ----- + ----- + ----- - 4 a z - ---- + ---- + 8 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 29 z 7 z 7 8 7 z 10 z 15 z 7 z 16 z > ----- + ---- - 9 a z - 12 z - ---- - ----- - ----- - ---- - ----- - 3 a 6 4 2 5 3 a a a a a a 9 10 10 9 z 3 z 3 z > ---- - ----- - ----- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 7 3 7 9 2 13 + 10 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 13 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 12 q t + 12 q t + 14 q t + 10 q t + 11 q t + 6 q t + 10 q t + 10 5 12 5 12 6 14 6 16 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a178

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