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DrawMorseLink

PD Presentation:

X12,1,13,2 X9,19,10,18 X5,14,6,15 X11,6,12,7 X15,11,16,22 X7,20,8,21 X3948 X21,17,22,16 X17,4,18,5 X10,13,1,14 X19,3,20,2

Gauss Code:

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

Jones Polynomial:

- 3q-9/2 + 7q-7/2 - 12q-5/2 + 14q-3/2 - 16q-1/2 + 15q1/2 - 12q3/2 + 8q5/2 - 4q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + 3q-14 - q-12 + 2q-10 + 4q-8 - 2q-6 + 4q-4 - 3q-2 + 1 - 2q4 + 3q6 - 2q8 + q10 + q12 - q14

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-4z2 + 2a-4z4 - a-4z6 + a-3z - 7a-3z3 + 10a-3z5 - 4a-3z7 - 7a-2z4 + 14a-2z6 - 6a-2z8 + 3a-1z - 16a-1z3 + 24a-1z5 - 4a-1z7 - 3a-1z9 + 3z2 - 16z4 + 31z6 - 14z8 + 4az - 18az3 + 27az5 - 8az7 - 3az9 + 3a2z2 - 10a2z4 + 13a2z6 - 8a2z8 - a3z-1 + 7a3z - 15a3z3 + 13a3z5 - 8a3z7 + a4 + a4z2 - 3a4z4 - 3a4z6 - a5z-1 + 5a5z - 6a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 3   j = 6               5 1   j = 4             7 3     j = 2           8 5       j = 0         8 7         j = -2       7 9           j = -4     5 7             j = -6   2 7               j = -8 1 5                 j = -10 3

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, NonAlternating, 249]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 249]]
Out[4]=	PD[X[12, 1, 13, 2], X[9, 19, 10, 18], X[5, 14, 6, 15], X[11, 6, 12, 7], > X[15, 11, 16, 22], X[7, 20, 8, 21], X[3, 9, 4, 8], X[21, 17, 22, 16], > X[17, 4, 18, 5], X[10, 13, 1, 14], X[19, 3, 20, 2]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -7, 9, -3, 4, -6, 7, -2, -10}, > {-4, -1, 10, 3, -5, 8, -9, 2, -11, 6, -8, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-3 7 12 14 16 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 12 q + 8 q - 4 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 3 -12 2 4 2 4 3 4 6 8 10 1 + q + --- - q + --- + -- - -- + -- - -- - 2 q + 3 q - 2 q + q + 14 10 8 6 4 2 q q q q q q 12 14 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 249]][a, z]
Out[8]=	3 5 3 3 a a z 3 z 3 5 z 5 z 3 3 3 -(--) + -- + -- - --- + 3 a z - 4 a z + a z + -- - ---- + 6 a z - 3 a z - z z 3 a 3 a a a 5 2 z 5 3 5 7 > ---- + 4 a z - a z + a z a
In[9]:=	Kauffman[Link[11, NonAlternating, 249]][a, z]
Out[9]=	3 5 2 4 a a z 3 z 3 5 2 z 2 2 a - -- - -- + -- + --- + 4 a z + 7 a z + 5 a z + 3 z - -- + 3 a z + z z 3 a 4 a a 3 3 4 4 4 2 7 z 16 z 3 3 3 5 3 4 2 z 7 z > a z - ---- - ----- - 18 a z - 15 a z - 6 a z - 16 z + ---- - ---- - 3 a 4 2 a a a 5 5 6 2 4 4 4 10 z 24 z 5 3 5 6 z > 10 a z - 3 a z + ----- + ----- + 27 a z + 13 a z + 31 z - -- + 3 a 4 a a 6 7 7 14 z 2 6 4 6 4 z 4 z 7 3 7 8 > ----- + 13 a z - 3 a z - ---- - ---- - 8 a z - 8 a z - 14 z - 2 3 a a a 8 9 6 z 2 8 3 z 9 > ---- - 8 a z - ---- - 3 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	9 3 1 5 2 7 5 7 7 8 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 7 t + 2 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 8 q t + 5 q t + 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 L11n249

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