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DrawMorseLink

PD Presentation:

X6172 X2,11,3,12 X14,3,15,4 X12,5,13,6 X22,13,5,14 X4,21,1,22 X20,16,21,15 X16,8,17,7 X18,10,19,9 X8,18,9,17 X10,20,11,19

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 3q-9/2 - 6q-7/2 + 9q-5/2 - 15q-3/2 + 16q-1/2 - 17q1/2 + 16q3/2 - 12q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-16 - q-14 + q-12 + 2q-10 + q-8 + 6q-6 + 2q-2 + 1 - 3q2 + 2q4 - 4q6 + q8 - q12 + 2q14 - q16

HOMFLY-PT Polynomial:

2a-3z3 + a-3z5 + a-1z-1 + 2a-1z - a-1z3 - 3a-1z5 - a-1z7 - 3az-1 - 6az - 6az3 - 4az5 - az7 + 2a3z-1 + 2a3z + 3a3z3 + a3z5

Kauffman Polynomial:

- a-6z4 + 2a-5z3 - 4a-5z5 - 2a-4z2 + 8a-4z4 - 8a-4z6 - 3a-3z3 + 12a-3z5 - 10a-3z7 - a-2 - 2a-2z2 + 11a-2z6 - 9a-2z8 + a-1z-1 - 2a-1z + 4a-1z3 - a-1z5 + 7a-1z7 - 6a-1z9 - 3 + 7z2 - 19z4 + 24z6 - 6z8 - 2z10 + 3az-1 - 8az + 24az3 - 39az5 + 33az7 - 10az9 - 3a2 + 10a2z2 - 23a2z4 + 17a2z6 - 2a2z10 + 2a3z-1 - 5a3z + 11a3z3 - 18a3z5 + 15a3z7 - 4a3z9 + 3a4z2 - 13a4z4 + 12a4z6 - 3a4z8 + a5z - 4a5z3 + 4a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   5 1   j = 6                 7 3     j = 4               9 5       j = 2             8 7         j = 0           9 10           j = -2         6 7             j = -4       3 9               j = -6     3 6                 j = -8   1 4                   j = -10   2                     j = -12 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, 137]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 137]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 11, 3, 12], X[14, 3, 15, 4], X[12, 5, 13, 6], > X[22, 13, 5, 14], X[4, 21, 1, 22], X[20, 16, 21, 15], X[16, 8, 17, 7], > X[18, 10, 19, 9], X[8, 18, 9, 17], X[10, 20, 11, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6}, {4, -1, 8, -10, 9, -11, 2, -4, 5, -3, 7, -8, 10, -9, > 11, -7, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 6 9 15 16 3/2 -q + ---- - ---- + ---- - ---- + ------- - 17 Sqrt[q] + 16 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 12 q + 8 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 2 -8 6 2 2 4 6 8 12 1 + q - q + q + --- + q + -- + -- - 3 q + 2 q - 4 q + q - q + 10 6 2 q q q 14 16 > 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 137]][a, z]
Out[8]=	3 3 3 5 1 3 a 2 a 2 z 3 2 z z 3 3 3 z --- - --- + ---- + --- - 6 a z + 2 a z + ---- - -- - 6 a z + 3 a z + -- - a z z z a 3 a 3 a a 5 7 3 z 5 3 5 z 7 > ---- - 4 a z + a z - -- - a z a a
In[9]:=	Kauffman[Link[11, Alternating, 137]][a, z]
Out[9]=	3 -2 2 1 3 a 2 a 2 z 3 5 2 -3 - a - 3 a + --- + --- + ---- - --- - 8 a z - 5 a z + a z + 7 z - a z z z a 2 2 3 3 3 2 z 2 z 2 2 4 2 2 z 3 z 4 z 3 > ---- - ---- + 10 a z + 3 a z + ---- - ---- + ---- + 24 a z + 4 2 5 3 a a a a a 4 4 5 3 3 5 3 4 z 8 z 2 4 4 4 4 z > 11 a z - 4 a z - 19 z - -- + ---- - 23 a z - 13 a z - ---- + 6 4 5 a a a 5 5 6 6 12 z z 5 3 5 5 5 6 8 z 11 z > ----- - -- - 39 a z - 18 a z + 4 a z + 24 z - ---- + ----- + 3 a 4 2 a a a 7 7 2 6 4 6 10 z 7 z 7 3 7 5 7 8 > 17 a z + 12 a z - ----- + ---- + 33 a z + 15 a z - a z - 6 z - 3 a a 8 9 9 z 4 8 6 z 9 3 9 10 2 10 > ---- - 3 a z - ---- - 10 a z - 4 a z - 2 z - 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 3 6 3 9 10 + 8 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + 12 6 10 5 8 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 6 9 7 2 4 4 2 6 2 6 3 > ----- + - + ---- + 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a137

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