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DrawMorseLink

PD Presentation:

X6172 X12,7,13,8 X4,13,1,14 X18,6,19,5 X8493 X20,9,21,10 X10,19,11,20 X22,16,17,15 X16,18,5,17 X14,22,15,21 X2,12,3,11

Gauss Code:

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

Jones Polynomial:

- q-4 + 3q-3 - 6q-2 + 11q-1 - 14 + 18q - 17q2 + 17q3 - 12q4 + 8q5 - 4q6 + q7

A2 (sl(3)) Invariant:

- q-12 + q-8 - q-6 + 3q-4 - q-2 + 3 + 5q2 + 2q4 + 9q6 + q8 + 5q10 + q12 - q14 + 2q16 - 2q18 + q20

HOMFLY-PT Polynomial:

a-4z-2 + a-4 + 2a-4z2 + 3a-4z4 + a-4z6 - 2a-2z-2 - 4a-2 - 8a-2z2 - 9a-2z4 - 5a-2z6 - a-2z8 + z-2 + 4 + 9z2 + 8z4 + 2z6 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + 2a-6z2 - 8a-6z4 + 8a-6z6 + a-5z3 - 11a-5z5 + 10a-5z7 - a-4z-2 + 3a-4 - 3a-4z2 + 6a-4z4 - 15a-4z6 + 10a-4z8 + 2a-3z-1 - 2a-3z + a-3z3 + a-3z5 - 10a-3z7 + 7a-3z9 - 2a-2z-2 + 7a-2 - 23a-2z2 + 48a-2z4 - 43a-2z6 + 10a-2z8 + 2a-2z10 + 2a-1z-1 - 4a-1z - 4a-1z3 + 29a-1z5 - 34a-1z7 + 11a-1z9 - z-2 + 7 - 27z2 + 48z4 - 32z6 + 3z8 + 2z10 - 3az + 2az3 + 9az5 - 13az7 + 4az9 + 2a2 - 9a2z2 + 15a2z4 - 12a2z6 + 3a2z8 - a3z + 4a3z3 - 4a3z5 + a3z7

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 = 15                       1 j = 13                     3   j = 11                   5 1   j = 9                 7 3     j = 7               10 5       j = 5             9 9         j = 3           9 8           j = 1         7 11             j = -1       4 7               j = -3     2 7                 j = -5   1 4                   j = -7   2                     j = -9 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, 472]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 472]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[18, 6, 19, 5], > X[8, 4, 9, 3], X[20, 9, 21, 10], X[10, 19, 11, 20], X[22, 16, 17, 15], > X[16, 18, 5, 17], X[14, 22, 15, 21], X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {9, -4, 7, -6, 10, -8}, > {4, -1, 2, -5, 6, -7, 11, -2, 3, -10, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 6 11 2 3 4 5 6 7 -14 - q + -- - -- + -- + 18 q - 17 q + 17 q - 12 q + 8 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -8 -6 3 -2 2 4 6 8 10 12 14 3 - q + q - q + -- - q + 5 q + 2 q + 9 q + q + 5 q + q - q + 4 q 16 18 20 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 472]][a, z]
Out[8]=	2 2 -4 4 2 -2 1 2 2 2 z 8 z 2 2 4 4 + a - -- - a + z + ----- - ----- + 9 z + ---- - ---- - 3 a z + 8 z + 2 4 2 2 2 4 2 a a z a z a a 4 4 6 6 8 3 z 9 z 2 4 6 z 5 z z > ---- - ---- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 472]][a, z]
Out[9]=	3 7 2 -2 1 2 2 2 2 z 4 z 7 + -- + -- + 2 a - z - ----- - ----- + ---- + --- - --- - --- - 3 a z - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 3 3 3 3 3 2 2 z 3 z 23 z 2 2 2 z z z 4 z > a z - 27 z + ---- - ---- - ----- - 9 a z - ---- + -- + -- - ---- + 6 4 2 7 5 3 a a a a a a a 4 4 4 4 5 3 3 3 4 z 8 z 6 z 48 z 2 4 4 z > 2 a z + 4 a z + 48 z + -- - ---- + ---- + ----- + 15 a z + ---- - 8 6 4 2 7 a a a a a 5 5 5 6 6 6 11 z z 29 z 5 3 5 6 8 z 15 z 43 z > ----- + -- + ----- + 9 a z - 4 a z - 32 z + ---- - ----- - ----- - 5 3 a 6 4 2 a a a a a 7 7 7 8 8 2 6 10 z 10 z 34 z 7 3 7 8 10 z 10 z > 12 a z + ----- - ----- - ----- - 13 a z + a z + 3 z + ----- + ----- + 5 3 a 4 2 a a a a 9 9 10 2 8 7 z 11 z 9 10 2 z > 3 a z + ---- + ----- + 4 a z + 2 z + ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 4 2 7 4 7 11 q + 9 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 7 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 8 q t + 9 q t + 9 q t + 10 q t + 5 q t + 7 q t + 3 q t + t 11 4 11 5 13 5 15 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a472

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