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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X18,5,19,6 X16,22,17,21 X20,16,21,15 X14,20,15,19 X2738 X4,11,5,12 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 7q-7/2 + 10q-5/2 - 15q-3/2 + 18q-1/2 - 18q1/2 + 17q3/2 - 13q5/2 + 8q7/2 - 4q9/2 + q11/2

A2 (sl(3)) Invariant:

q-18 + 2q-16 + q-14 + 4q-12 + 3q-10 + 4q-6 - 3q-4 - q-2 - 1 - 4q2 + 3q4 - 3q6 + 3q8 + q10 - 2q12 + 2q14 - q16

HOMFLY-PT Polynomial:

a-3z + 2a-3z3 + a-3z5 - 5a-1z - 7a-1z3 - 4a-1z5 - a-1z7 + az-1 + 7az + 9az3 + 3az5 - 3a3z-1 - 7a3z - 3a3z3 + 2a5z-1 + a5z

Kauffman Polynomial:

- a-6z4 + 2a-5z3 - 4a-5z5 - 2a-4z2 + 7a-4z4 - 8a-4z6 + a-3z - 6a-3z3 + 14a-3z5 - 11a-3z7 - a-2z2 - a-2z4 + 12a-2z6 - 10a-2z8 + 3a-1z - 23a-1z3 + 32a-1z5 - 6a-1z7 - 5a-1z9 + 1 + 13z2 - 41z4 + 48z6 - 16z8 - z10 - az-1 - 9az3 + 5az5 + 14az7 - 8az9 + 3a2 + 8a2z2 - 35a2z4 + 34a2z6 - 8a2z8 - a2z10 - 3a3z-1 + 5a3z - 3a3z3 - 4a3z5 + 8a3z7 - 3a3z9 + 3a4 - 4a4z2 - 3a4z4 + 6a4z6 - 2a4z8 - 2a5z-1 + 7a5z - 9a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     3   j = 8                   5 1   j = 6                 8 3     j = 4               9 5       j = 2             9 8         j = 0           10 10           j = -2         5 8             j = -4       5 10               j = -6     2 5                 j = -8     5                   j = -10 1 2                     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, 240]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 240]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[22, 10, 7, 9], X[10, 14, 11, 13], > X[18, 5, 19, 6], X[16, 22, 17, 21], X[20, 16, 21, 15], X[14, 20, 15, 19], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 5, -11}, > {9, -1, 3, -4, 10, -2, 4, -8, 7, -6, 11, -5, 8, -7, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 7 10 15 18 3/2 -q + ---- - ---- + ---- - ---- + ------- - 18 Sqrt[q] + 17 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]=	-18 2 -14 4 3 4 3 -2 2 4 6 -1 + q + --- + q + --- + --- + -- - -- - q - 4 q + 3 q - 3 q + 16 12 10 6 4 q q q q q 8 10 12 14 16 > 3 q + q - 2 q + 2 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 240]][a, z]
Out[8]=	3 5 3 3 a 3 a 2 a z 5 z 3 5 2 z 7 z 3 - - ---- + ---- + -- - --- + 7 a z - 7 a z + a z + ---- - ---- + 9 a z - z z z 3 a 3 a a a 5 5 7 3 3 z 4 z 5 z > 3 a z + -- - ---- + 3 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 240]][a, z]
Out[9]=	3 5 2 2 4 a 3 a 2 a z 3 z 3 5 2 2 z 1 + 3 a + 3 a - - - ---- - ---- + -- + --- + 5 a z + 7 a z + 13 z - ---- - z z z 3 a 4 a a 2 3 3 3 z 2 2 4 2 2 z 6 z 23 z 3 3 3 5 3 > -- + 8 a z - 4 a z + ---- - ---- - ----- - 9 a z - 3 a z - 9 a z - 2 5 3 a a a a 4 4 4 5 5 5 4 z 7 z z 2 4 4 4 4 z 14 z 32 z > 41 z - -- + ---- - -- - 35 a z - 3 a z - ---- + ----- + ----- + 6 4 2 5 3 a a a a a a 6 6 5 3 5 5 5 6 8 z 12 z 2 6 4 6 > 5 a z - 4 a z + 5 a z + 48 z - ---- + ----- + 34 a z + 6 a z - 4 2 a a 7 7 8 11 z 6 z 7 3 7 5 7 8 10 z 2 8 > ----- - ---- + 14 a z + 8 a z - a z - 16 z - ----- - 8 a z - 3 a 2 a a 9 4 8 5 z 9 3 9 10 2 10 > 2 a z - ---- - 8 a z - 3 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 5 2 5 5 10 10 + 9 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 6 10 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 5 10 8 2 4 4 2 6 2 6 3 > ----- + -- + ---- + 8 q t + 9 q t + 5 q t + 8 q t + 3 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a240

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