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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 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 8q-5/2 - 14q-3/2 + 18q-1/2 - 22q1/2 + 21q3/2 - 19q5/2 + 14q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-12 + 2q-10 - 2q-8 + 4q-6 + q-4 + q-2 + 6 - 3q2 + 6q4 - 3q6 + q8 - 4q12 + 2q14 - 2q16 + q18

HOMFLY-PT Polynomial:

a-3z-1 + a-3z - 4a-3z3 - 4a-3z5 - a-3z7 - 3a-1z-1 - 3a-1z + 7a-1z3 + 12a-1z5 + 6a-1z7 + a-1z9 + 2az-1 + az - 4az3 - 4az5 - az7

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 6a-6z4 - 4a-6z6 - 3a-5z3 + 10a-5z5 - 7a-5z7 + a-4 - 5a-4z2 + 2a-4z4 + 9a-4z6 - 8a-4z8 - a-3z-1 + a-3z + 3a-3z3 - 6a-3z5 + 9a-3z7 - 7a-3z9 + 3a-2 - 3a-2z2 - 10a-2z4 + 16a-2z6 - 5a-2z8 - 3a-2z10 - 3a-1z-1 + 4a-1z + 14a-1z3 - 38a-1z5 + 36a-1z7 - 14a-1z9 + 3 + 2z2 - 20z4 + 22z6 - 4z8 - 3z10 - 2az-1 + 3az + 2az3 - 11az5 + 16az7 - 7az9 + 2a2z2 - 12a2z4 + 18a2z6 - 7a2z8 - 5a3z3 + 10a3z5 - 4a3z7 + 2a4z4 - a4z6

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 = 14                       1 j = 12                     3   j = 10                   5 1   j = 8                 9 3     j = 6               10 5       j = 4             11 9         j = 2           11 10           j = 0         8 12             j = -2       6 10               j = -4     3 9                 j = -6   1 5                   j = -8   3                     j = -10 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, 141]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 141]]
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, 14, 21, 13], > X[22, 16, 7, 15], X[4, 20, 5, 19], X[14, 22, 15, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10, 5, -6}, > {6, -1, 7, -3, 2, -5, 8, -11, 9, -7, 4, -2, 10, -8, 11, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 8 14 18 3/2 5/2 q - ---- + ---- - ---- + ------- - 22 Sqrt[q] + 21 q - 19 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 14 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 2 2 4 -4 -2 2 4 6 8 12 6 - q + --- - -- + -- + q + q - 3 q + 6 q - 3 q + q - 4 q + 10 8 6 q q q 14 16 18 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 141]][a, z]
Out[8]=	3 3 5 5 1 3 2 a z 3 z 4 z 7 z 3 4 z 12 z ---- - --- + --- + -- - --- + a z - ---- + ---- - 4 a z - ---- + ----- - 3 a z z 3 a 3 a 3 a a z a a a 7 7 9 5 z 6 z 7 z > 4 a z - -- + ---- - a z + -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 141]][a, z]
Out[9]=	2 2 -4 3 1 3 2 a z 4 z 2 2 z 5 z 3 + a + -- - ---- - --- - --- + -- + --- + 3 a z + 2 z - ---- - ---- - 2 3 a z z 3 a 6 4 a a z a a a 2 3 3 3 3 3 z 2 2 z 3 z 3 z 14 z 3 3 3 4 > ---- + 2 a z + -- - ---- + ---- + ----- + 2 a z - 5 a z - 20 z + 2 7 5 3 a a a a a 4 4 4 5 5 5 5 6 z 2 z 10 z 2 4 4 4 z 10 z 6 z 38 z > ---- + ---- - ----- - 12 a z + 2 a z - -- + ----- - ---- - ----- - 6 4 2 7 5 3 a a a a a a a 6 6 6 5 3 5 6 4 z 9 z 16 z 2 6 4 6 > 11 a z + 10 a z + 22 z - ---- + ---- + ----- + 18 a z - a z - 6 4 2 a a a 7 7 7 8 8 7 z 9 z 36 z 7 3 7 8 8 z 5 z 2 8 > ---- + ---- + ----- + 16 a z - 4 a z - 4 z - ---- - ---- - 7 a z - 5 3 a 4 2 a a a a 9 9 10 7 z 14 z 9 10 3 z > ---- - ----- - 7 a z - 3 z - ----- 3 a 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 9 6 8 12 + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 10 2 4 4 2 6 2 6 3 8 3 > ---- + 10 q t + 11 q t + 9 q t + 10 q t + 5 q t + 9 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a141

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