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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 6q-11/2 - 11q-9/2 + 14q-7/2 - 17q-5/2 + 17q-3/2 - 15q-1/2 + 11q1/2 - 7q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + q-20 - q-18 + q-16 + 4q-14 - q-12 + 4q-10 - q-8 - q-6 + q-4 - 3q-2 + 4 - q2 + q4 + 2q6 - q8 + q10

HOMFLY-PT Polynomial:

- a-1z-1 - 3a-1z - 3a-1z3 - a-1z5 + 2az-1 + 4az + 6az3 + 4az5 + az7 - 2a3z-1 + 5a3z3 + 4a3z5 + a3z7 + a5z-1 - 2a5z - 3a5z3 - a5z5

Kauffman Polynomial:

2a-3z3 - a-3z5 + 5a-2z4 - 3a-2z6 - a-1z-1 + 5a-1z - 12a-1z3 + 14a-1z5 - 6a-1z7 + 6z2 - 18z4 + 17z6 - 7z8 - 2az-1 + 8az - 11az3 - 2az5 + 8az7 - 5az9 - a2 + 7a2z2 - 19a2z4 + 14a2z6 - 3a2z8 - 2a2z10 - 2a3z-1 + 2a3z + 18a3z3 - 35a3z5 + 26a3z7 - 9a3z9 + 2a4z2 - 2a4z4 + 4a4z6 - 2a4z10 - a5z-1 + 8a5z3 - 9a5z5 + 9a5z7 - 4a5z9 - a6z2 - 3a6z4 + 9a6z6 - 4a6z8 + a7z - 7a7z3 + 9a7z5 - 3a7z7 - 2a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     2   j = 4                   5 1   j = 2                 6 2     j = 0               9 5       j = -2             9 7         j = -4           8 8           j = -6         7 10             j = -8       4 7               j = -10     2 7                 j = -12   1 4                   j = -14   2                     j = -16 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, 174]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 174]]
Out[4]=	PD[X[8, 1, 9, 2], X[2, 9, 3, 10], X[10, 3, 11, 4], X[16, 8, 17, 7], > X[22, 16, 7, 15], X[6, 21, 1, 22], X[20, 11, 21, 12], X[14, 6, 15, 5], > X[18, 13, 19, 14], X[12, 19, 13, 20], X[4, 18, 5, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 8, -6}, > {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -9, 10, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 6 11 14 17 17 15 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 11 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 7 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 -16 4 -12 4 -8 -6 -4 3 2 4 - q + q - q + q + --- - q + --- - q - q + q - -- - q + 14 10 2 q q q 4 6 8 10 > q + 2 q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 174]][a, z]
Out[8]=	3 5 3 1 2 a 2 a a 3 z 5 3 z 3 3 3 -(---) + --- - ---- + -- - --- + 4 a z - 2 a z - ---- + 6 a z + 5 a z - a z z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 3 a z - -- + 4 a z + 4 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 174]][a, z]
Out[9]=	3 5 2 1 2 a 2 a a 5 z 3 7 2 2 2 -a - --- - --- - ---- - -- + --- + 8 a z + 2 a z + a z + 6 z + 7 a z + a z z z z a 3 3 4 2 6 2 8 2 2 z 12 z 3 3 3 5 3 > 2 a z - a z - 2 a z + ---- - ----- - 11 a z + 18 a z + 8 a z - 3 a a 4 5 7 3 4 5 z 2 4 4 4 6 4 8 4 z > 7 a z - 18 z + ---- - 19 a z - 2 a z - 3 a z + 3 a z - -- + 2 3 a a 5 6 14 z 5 3 5 5 5 7 5 6 3 z 2 6 > ----- - 2 a z - 35 a z - 9 a z + 9 a z + 17 z - ---- + 14 a z + a 2 a 7 4 6 6 6 8 6 6 z 7 3 7 5 7 7 7 > 4 a z + 9 a z - a z - ---- + 8 a z + 26 a z + 9 a z - 3 a z - a 8 2 8 6 8 9 3 9 5 9 2 10 4 10 > 7 z - 3 a z - 4 a z - 5 a z - 9 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 2 1 4 2 7 4 7 9 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 10 8 8 9 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 5 t + 6 q t + 2 q t + 5 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a174

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