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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X22,14,11,13 X20,16,21,15 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,20,5,19 X14,22,15,21 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-3 - q-2 + 4q-1 - 5 + 9q - 10q2 + 11q3 - 10q4 + 8q5 - 5q6 + 3q7 - q8

A2 (sl(3)) Invariant:

q-10 + 2q-8 + 2q-6 + 4q-4 + 3q-2 + 3 + 5q2 + q4 + 4q6 + q8 + q12 - 2q14 + 2q16 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6z2 - a-6z4 + 2a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 + 3a-2 + 3a-2z2 + 3a-2z4 + a-2z6 - 2z-2 - 6 - 7z2 - 2z4 + a2z-2 + 3a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 2a-8z2 - 7a-8z4 + 3a-8z6 + 4a-7z3 - 9a-7z5 + 4a-7z7 - 2a-6z2 + 9a-6z4 - 10a-6z6 + 4a-6z8 - 6a-5z3 + 14a-5z5 - 9a-5z7 + 3a-5z9 + a-4 - 4a-4z2 + 10a-4z4 - 4a-4z6 + a-4z10 - 14a-3z3 + 26a-3z5 - 15a-3z7 + 4a-3z9 + a-2z-2 - 3a-2 + 5a-2z2 - 10a-2z4 + 8a-2z6 - 3a-2z8 + a-2z10 - 2a-1z-1 + 6a-1z - 5a-1z3 - a-1z7 + a-1z9 + 2z-2 - 8 + 13z2 - 9z4 + z8 - 2az-1 + 6az - 3az3 - 2az5 + az7 + a2z-2 - 5a2 + 8a2z2 - 5a2z4 + a2z6

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   3 1   j = 11                 5 2     j = 9               5 3       j = 7             6 5         j = 5           6 7           j = 3         3 4             j = 1       3 7               j = -1     1 2                 j = -3     3                   j = -5 1 1                     j = -7 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 417]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 417]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[22, 14, 11, 13], X[20, 16, 21, 15], > X[8, 18, 9, 17], X[16, 8, 17, 7], X[18, 10, 19, 9], X[10, 20, 5, 19], > X[14, 22, 15, 21], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 6, -5, 7, -8}, > {11, -2, 3, -9, 4, -6, 5, -7, 8, -4, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-3 -2 4 2 3 4 5 6 7 8 -5 + q - q + - + 9 q - 10 q + 11 q - 10 q + 8 q - 5 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 4 3 2 4 6 8 12 14 16 3 + q + -- + -- + -- + -- + 5 q + q + 4 q + q + q - 2 q + 2 q + 8 6 4 2 q q q q 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 417]][a, z]
Out[8]=	2 2 2 2 3 2 2 1 a 2 2 z 2 z 3 z 2 2 4 -6 + -- + 3 a - -- + ----- + -- - 7 z - ---- + ---- + ---- + a z - 2 z - 2 2 2 2 2 6 4 2 a z a z z a a a 4 4 4 6 6 z 3 z 3 z z z > -- + ---- + ---- + -- + -- 6 4 2 4 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 417]][a, z]
Out[9]=	2 -4 3 2 2 1 a 2 2 a 6 z 2 -8 + a - -- - 5 a + -- + ----- + -- - --- - --- + --- + 6 a z + 13 z + 2 2 2 2 2 a z z a a z a z z 2 2 2 2 3 3 3 3 3 2 z 2 z 4 z 5 z 2 2 2 z 4 z 6 z 14 z 5 z > ---- - ---- - ---- + ---- + 8 a z - ---- + ---- - ---- - ----- - ---- - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 3 4 7 z 9 z 10 z 10 z 2 4 z 9 z 14 z > 3 a z - 9 z - ---- + ---- + ----- - ----- - 5 a z + -- - ---- + ----- + 8 6 4 2 9 7 5 a a a a a a a 5 6 6 6 6 7 7 7 26 z 5 3 z 10 z 4 z 8 z 2 6 4 z 9 z 15 z > ----- - 2 a z + ---- - ----- - ---- + ---- + a z + ---- - ---- - ----- - 3 8 6 4 2 7 5 3 a a a a a a a a 7 8 8 9 9 9 10 10 z 7 8 4 z 3 z 3 z 4 z z z z > -- + a z + z + ---- - ---- + ---- + ---- + -- + --- + --- a 6 2 5 3 a 4 2 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 3 1 2 3 q 3 7 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 4 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 6 q t + 7 q t + 6 q t + 5 q t + 5 q t + 3 q t + 5 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a417

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