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DrawMorseLink

PD Presentation:

X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19

Gauss Code:

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

Jones Polynomial:

q-11/2 - 4q-9/2 + 8q-7/2 - 14q-5/2 + 18q-3/2 - 21q-1/2 + 20q1/2 - 18q3/2 + 13q5/2 - 8q7/2 + 4q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-16 + 2q-14 - q-12 + q-10 + 4q-8 - 2q-6 + 5q-4 - q-2 + 1 + 2q2 - 3q4 + 4q6 - 2q8 + q12 - 2q14 + q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-5z3 + 3a-5z5 - a-5z7 + 4a-4z2 - 14a-4z4 + 14a-4z6 - 4a-4z8 + 6a-3z3 - 19a-3z5 + 20a-3z7 - 6a-3z9 + 12a-2z2 - 37a-2z4 + 31a-2z6 - 2a-2z8 - 3a-2z10 + 8a-1z3 - 35a-1z5 + 44a-1z7 - 15a-1z9 + 12z2 - 44z4 + 48z6 - 11z8 - 3z10 + az-1 - az - 8az3 + 7az5 + 11az7 - 9az9 - a2 + 4a2z2 - 14a2z4 + 23a2z6 - 13a2z8 + a3z-1 - a3z - 6a3z3 + 16a3z5 - 12a3z7 + 6a4z4 - 8a4z6 + 2a5z3 - 4a5z5 - 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                     3   j = 8                   5 1   j = 6                 8 3     j = 4               10 5       j = 2             10 8         j = 0           11 10           j = -2         8 11             j = -4       6 10               j = -6     3 9                 j = -8   1 5                   j = -10   3                     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, 142]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 142]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 11, 19, 12], X[10, 4, 11, 3], X[2, 17, 3, 18], > X[12, 5, 13, 6], X[6, 7, 1, 8], X[16, 10, 17, 9], X[20, 16, 21, 15], > X[22, 14, 7, 13], X[14, 22, 15, 21], X[4, 20, 5, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11, 5, -6}, > {6, -1, 7, -3, 2, -5, 9, -10, 8, -7, 4, -2, 11, -8, 10, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 4 8 14 18 21 3/2 q - ---- + ---- - ---- + ---- - ------- + 20 Sqrt[q] - 18 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 13 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 -12 -10 4 2 5 -2 2 4 6 8 1 - q + --- - q + q + -- - -- + -- - q + 2 q - 3 q + 4 q - 2 q + 14 8 6 4 q q q q 12 14 16 > q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 142]][a, z]
Out[8]=	3 3 3 5 5 a a 2 z 2 z 3 3 3 z 3 z 5 3 5 -(-) + -- - a z - ---- + ---- + 2 a z - 2 a z - -- + ---- + 3 a z - a z + z z 3 a 3 a a a 7 z 7 > -- + a z a
In[9]:=	Kauffman[Link[11, Alternating, 142]][a, z]
Out[9]=	3 2 2 3 3 2 a a 3 2 4 z 12 z 2 2 2 z 6 z -a + - + -- - a z - a z + 12 z + ---- + ----- + 4 a z - ---- + ---- + z z 4 2 5 3 a a a a 3 4 4 8 z 3 3 3 5 3 4 14 z 37 z 2 4 > ---- - 8 a z - 6 a z + 2 a z - 44 z - ----- - ----- - 14 a z + a 4 2 a a 5 5 5 4 4 6 4 3 z 19 z 35 z 5 3 5 5 5 > 6 a z - a z + ---- - ----- - ----- + 7 a z + 16 a z - 4 a z + 5 3 a a a 6 6 7 7 7 6 14 z 31 z 2 6 4 6 z 20 z 44 z 7 > 48 z + ----- + ----- + 23 a z - 8 a z - -- + ----- + ----- + 11 a z - 4 2 5 3 a a a a a 8 8 9 9 3 7 8 4 z 2 z 2 8 6 z 15 z 9 10 > 12 a z - 11 z - ---- - ---- - 13 a z - ---- - ----- - 9 a z - 3 z - 4 2 3 a a a a 10 3 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 5 3 9 6 10 11 + -- + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 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 8 2 2 2 4 2 4 3 6 3 6 4 > ---- + 10 t + 10 q t + 8 q t + 10 q t + 5 q t + 8 q t + 3 q t + 2 q t 8 4 8 5 10 5 12 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a142

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