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DrawMorseLink

PD Presentation:

X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X22,13,9,14 X14,6,15,5 X4,22,5,21 X18,8,19,7 X20,16,21,15 X16,20,17,19 X6,18,7,17

Gauss Code:

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

Jones Polynomial:

q-11/2 - 3q-9/2 + 6q-7/2 - 10q-5/2 + 12q-3/2 - 14q-1/2 + 13q1/2 - 12q3/2 + 8q5/2 - 5q7/2 + 3q9/2 - q11/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-3z - 3a-3z3 - a-3z5 - a-1z-1 + 4a-1z3 + 4a-1z5 + a-1z7 + az-1 + 3az + 5az3 + 4az5 + az7 - 2a3z - 3a3z3 - a3z5

Kauffman Polynomial:

a-5z - 4a-5z3 + 4a-5z5 - a-5z7 + 5a-4z2 - 16a-4z4 + 13a-4z6 - 3a-4z8 + a-3z - a-3z3 - 7a-3z5 + 10a-3z7 - 3a-3z9 + 9a-2z2 - 28a-2z4 + 24a-2z6 - 4a-2z8 - a-2z10 - a-1z-1 + 2a-1z + a-1z3 - 10a-1z5 + 16a-1z7 - 6a-1z9 + 1 + 5z2 - 16z4 + 20z6 - 6z8 - z10 - az-1 + 4az - 9az3 + 11az5 - az7 - 3az9 - 2a2z2 + 2a2z4 + 4a2z6 - 5a2z8 + a3z - 4a3z3 + 7a3z5 - 6a3z7 - 2a4z2 + 5a4z4 - 5a4z6 - a5z + 3a5z3 - 3a5z5 + a6z2 - a6z4

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                       1 j = 10                     2   j = 8                   3 1   j = 6                 5 2     j = 4               7 3       j = 2             6 5         j = 0           8 7           j = -2         6 8             j = -4       4 6               j = -6     2 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, 280]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 280]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[8, 9, 1, 10], > X[22, 13, 9, 14], X[14, 6, 15, 5], X[4, 22, 5, 21], X[18, 8, 19, 7], > X[20, 16, 21, 15], X[16, 20, 17, 19], X[6, 18, 7, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -7, 6, -11, 8, -4}, > {4, -1, 2, -3, 5, -6, 9, -10, 11, -8, 10, -9, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 3 6 10 12 14 3/2 q - ---- + ---- - ---- + ---- - ------- + 13 Sqrt[q] - 12 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 8 q - 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -14 -12 2 -6 4 2 4 6 8 14 16 2 - q + q - q + -- - q + -- + 2 q - q + 3 q - q - q + q 8 4 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 280]][a, z]
Out[8]=	3 3 5 5 1 a z 3 3 z 4 z 3 3 3 z 4 z -(---) + - - -- + 3 a z - 2 a z - ---- + ---- + 5 a z - 3 a z - -- + ---- + a z z 3 3 a 3 a a a a 7 5 3 5 z 7 > 4 a z - a z + -- + a z a
In[9]:=	Kauffman[Link[11, Alternating, 280]][a, z]
Out[9]=	2 2 1 a z z 2 z 3 5 2 5 z 9 z 1 - --- - - + -- + -- + --- + 4 a z + a z - a z + 5 z + ---- + ---- - a z z 5 3 a 4 2 a a a a 3 3 3 2 2 4 2 6 2 4 z z z 3 3 3 5 3 > 2 a z - 2 a z + a z - ---- - -- + -- - 9 a z - 4 a z + 3 a z - 5 3 a a a 4 4 5 5 5 4 16 z 28 z 2 4 4 4 6 4 4 z 7 z 10 z > 16 z - ----- - ----- + 2 a z + 5 a z - a z + ---- - ---- - ----- + 4 2 5 3 a a a a a 6 6 5 3 5 5 5 6 13 z 24 z 2 6 4 6 > 11 a z + 7 a z - 3 a z + 20 z + ----- + ----- + 4 a z - 5 a z - 4 2 a a 7 7 7 8 8 9 z 10 z 16 z 7 3 7 8 3 z 4 z 2 8 3 z > -- + ----- + ----- - a z - 6 a z - 6 z - ---- - ---- - 5 a z - ---- - 5 3 a 4 2 3 a a a a a 9 10 6 z 9 10 z > ---- - 3 a z - z - --- a 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 4 2 6 4 6 8 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 2 12 5 10 4 8 4 8 3 6 3 6 2 4 2 4 q q t q t q t q t q t q t q t q t 6 2 2 2 4 2 4 3 6 3 6 4 > ---- + 7 t + 6 q t + 5 q t + 7 q t + 3 q t + 5 q t + 2 q t + 2 q t 8 4 8 5 10 5 12 6 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a280

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