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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,14,21,13 X14,5,15,6 X4,19,5,20 X18,15,19,16 X16,8,17,7 X6,18,7,17 X8,22,1,21

Gauss Code:

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

Jones Polynomial:

q-9/2 - 4q-7/2 + 8q-5/2 - 13q-3/2 + 17q-1/2 - 21q1/2 + 20q3/2 - 18q5/2 + 13q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 + q-10 - 2q-8 + 4q-6 - q-4 + 4 - 2q2 + 5q4 - q6 + 2q8 + 2q10 - 4q12 + 2q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + 4a-3z + 5a-3z3 + 2a-3z5 - a-1z-1 - 6a-1z - 8a-1z3 - 4a-1z5 - a-1z7 + az-1 + 4az + 5az3 + 2az5 - a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - 2a-6z2 + 6a-6z4 - 4a-6z6 + 2a-5z - 6a-5z3 + 11a-5z5 - 7a-5z7 - 3a-4z2 + 3a-4z4 + 6a-4z6 - 7a-4z8 + 8a-3z - 25a-3z3 + 28a-3z5 - 8a-3z7 - 4a-3z9 + 2a-2z2 - 20a-2z4 + 32a-2z6 - 15a-2z8 - a-2z10 - a-1z-1 + 12a-1z - 35a-1z3 + 30a-1z5 + a-1z7 - 8a-1z9 + 1 + 4z2 - 25z4 + 36z6 - 14z8 - z10 - az-1 + 8az - 24az3 + 24az5 - 2az7 - 4az9 - 6a2z4 + 13a2z6 - 6a2z8 + 2a3z - 7a3z3 + 10a3z5 - 4a3z7 - a4z2 + 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                 8 3     j = 6               10 5       j = 4             10 8         j = 2           11 10           j = 0         8 12             j = -2       5 9               j = -4     3 8                 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, 247]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 247]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[2, 9, 3, 10], > X[20, 14, 21, 13], X[14, 5, 15, 6], X[4, 19, 5, 20], X[18, 15, 19, 16], > X[16, 8, 17, 7], X[6, 18, 7, 17], X[8, 22, 1, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, -7, 6, -10, 9, -11}, > {4, -1, 3, -2, 5, -6, 8, -9, 10, -8, 7, -5, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 4 8 13 17 3/2 5/2 q - ---- + ---- - ---- + ------- - 21 Sqrt[q] + 20 q - 18 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 13 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 4 -4 2 4 6 8 10 4 - q + q + q - -- + -- - q - 2 q + 5 q - q + 2 q + 2 q - 8 6 q q 12 14 16 18 20 > 4 q + 2 q - q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 247]][a, z]
Out[8]=	3 3 3 1 a z 4 z 6 z 3 z 5 z 8 z 3 -(---) + - - -- + --- - --- + 4 a z - a z - -- + ---- - ---- + 5 a z - a z z 5 3 a 5 3 a a a a a 5 5 7 3 3 2 z 4 z 5 z > a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 247]][a, z]
Out[9]=	2 2 2 1 a 2 z 8 z 12 z 3 2 2 z 3 z 2 z 1 - --- - - + --- + --- + ---- + 8 a z + 2 a z + 4 z - ---- - ---- + ---- - a z z 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 2 z 6 z 25 z 35 z 3 3 3 4 6 z > a z + -- - ---- - ----- - ----- - 24 a z - 7 a z - 25 z + ---- + 7 5 3 a 6 a a a a 4 4 5 5 5 5 3 z 20 z 2 4 4 4 z 11 z 28 z 30 z 5 > ---- - ----- - 6 a z + 2 a z - -- + ----- + ----- + ----- + 24 a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 3 5 6 4 z 6 z 32 z 2 6 4 6 7 z 8 z > 10 a z + 36 z - ---- + ---- + ----- + 13 a z - a z - ---- - ---- + 6 4 2 5 3 a a a a a 7 8 8 9 9 z 7 3 7 8 7 z 15 z 2 8 4 z 8 z > -- - 2 a z - 4 a z - 14 z - ---- - ----- - 6 a z - ---- - ---- - a 4 2 3 a a a a 10 9 10 z > 4 a z - z - --- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 3 1 5 3 8 5 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 9 2 4 4 2 6 2 6 3 8 3 > ---- + 10 q t + 10 q t + 8 q t + 10 q t + 5 q t + 8 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 L11a247

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