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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-13/2 - 3q-11/2 + 4q-9/2 - 7q-7/2 + 9q-5/2 - 10q-3/2 + 11q-1/2 - 10q1/2 + 7q3/2 - 6q5/2 + 3q7/2 - q9/2

A2 (sl(3)) Invariant:

- q-20 + q-18 + q-16 + q-14 + 4q-12 - q-8 - q-6 - 3q-4 + q2 + 3q4 + 3q8 + q10 - q12 + q14

HOMFLY-PT Polynomial:

- a-3z-1 - a-3z - a-3z3 + a-1z-1 + 2a-1z + 2a-1z3 + a-1z5 + az-1 + az3 + az5 - 2a3z-1 + 2a3z3 + a3z5 + a5z-1 - a5z - a5z3

Kauffman Polynomial:

- a-5z3 - 3a-4z4 + a-3z-1 - 3a-3z + 6a-3z3 - 6a-3z5 - a-2 + 8a-2z4 - 7a-2z6 + a-1z-1 - 3a-1z + 3a-1z3 + 10a-1z5 - 7a-1z7 - 2 + 7z2 - 12z4 + 18z6 - 7z8 - az-1 + 6az - 6az3 - 5az5 + 14az7 - 5az9 - 3a2 + 10a2z2 - 29a2z4 + 20a2z6 + a2z8 - 2a2z10 - 2a3z-1 + 7a3z + 15a3z3 - 51a3z5 + 38a3z7 - 8a3z9 - a4 + 6a4z2 - 13a4z4 + 7a4z8 - 2a4z10 - a5z-1 + a5z + 17a5z3 - 30a5z5 + 17a5z7 - 3a5z9 + 3a6z2 - 7a6z4 + 5a6z6 - a6z8

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 = 10                       1 j = 8                     2   j = 6                   4 1   j = 4                 3 2     j = 2               7 4       j = 0             6 5         j = -2           4 5           j = -4         5 6             j = -6       2 4               j = -8     2 5                 j = -10   1 2                   j = -12   2                     j = -14 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, 103]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 103]]
Out[4]=	PD[X[6, 1, 7, 2], X[20, 7, 21, 8], X[4, 21, 1, 22], X[14, 6, 15, 5], > X[12, 4, 13, 3], X[22, 14, 5, 13], X[18, 9, 19, 10], X[16, 11, 17, 12], > X[2, 16, 3, 15], X[10, 17, 11, 18], X[8, 19, 9, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {4, -1, 2, -11, 7, -10, 8, -5, 6, -4, 9, -8, 10, -7, > 11, -2, 3, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 3 4 7 9 10 11 3/2 q - ----- + ---- - ---- + ---- - ---- + ------- - 10 Sqrt[q] + 7 q - 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 9/2 > 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -18 -16 -14 4 -8 -6 3 2 4 8 10 -q + q + q + q + --- - q - q - -- + q + 3 q + 3 q + q - 12 4 q q 12 14 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 103]][a, z]
Out[8]=	3 5 3 3 1 1 a 2 a a z 2 z 5 z 2 z 3 3 3 -(----) + --- + - - ---- + -- - -- + --- - a z - -- + ---- + a z + 2 a z - 3 a z z z z 3 a 3 a a z a a 5 5 3 z 5 3 5 > a z + -- + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 103]][a, z]
Out[9]=	3 5 -2 2 4 1 1 a 2 a a 3 z 3 z -2 - a - 3 a - a + ---- + --- - - - ---- - -- - --- - --- + 6 a z + 3 a z z z z 3 a a z a 3 3 3 3 5 2 2 2 4 2 6 2 z 6 z 3 z > 7 a z + a z + 7 z + 10 a z + 6 a z + 3 a z - -- + ---- + ---- - 5 3 a a a 4 4 3 3 3 5 3 4 3 z 8 z 2 4 4 4 > 6 a z + 15 a z + 17 a z - 12 z - ---- + ---- - 29 a z - 13 a z - 4 2 a a 5 5 6 6 4 6 z 10 z 5 3 5 5 5 6 7 z > 7 a z - ---- + ----- - 5 a z - 51 a z - 30 a z + 18 z - ---- + 3 a 2 a a 7 2 6 6 6 7 z 7 3 7 5 7 8 2 8 > 20 a z + 5 a z - ---- + 14 a z + 38 a z + 17 a z - 7 z + a z + a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 7 a z - a z - 5 a z - 8 a z - 3 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 2 2 5 2 4 5 + 7 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 14 7 12 6 10 6 10 5 8 5 8 4 6 4 6 3 q t q t q t q t q t q t q t q t 5 6 4 6 5 2 4 4 2 6 2 > ----- + ----- + ----- + - + ---- + 4 q t + 3 q t + 2 q t + 4 q t + 4 3 4 2 2 2 t 2 q t q t q t q t 6 3 8 3 10 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a103

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