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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 6q-5/2 - 11q-3/2 + 14q-1/2 - 18q1/2 + 18q3/2 - 16q5/2 + 12q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 + q-12 - 2q-8 + 4q-6 + q-2 + 5 - q2 + 3q4 - 2q6 + 2q10 - 3q12 + 3q14 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-5z3 - a-3z + a-3z5 - a-1z-1 + a-1z + 3a-1z3 + 2a-1z5 + az-1 + az + az3 + az5 - a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - a-6z2 + 6a-6z4 - 4a-6z6 + a-5z - 5a-5z3 + 12a-5z5 - 7a-5z7 + a-4z2 - 4a-4z4 + 10a-4z6 - 7a-4z8 + 2a-3z - 10a-3z3 + 13a-3z5 - 2a-3z7 - 4a-3z9 + 5a-2z2 - 23a-2z4 + 27a-2z6 - 11a-2z8 - a-2z10 - a-1z-1 + 4a-1z - 9a-1z3 + 2a-1z5 + 9a-1z7 - 7a-1z9 + 1 + 5z2 - 20z4 + 23z6 - 8z8 - z10 - az-1 + 6az - 13az3 + 11az5 + az7 - 3az9 - 4a2z4 + 9a2z6 - 4a2z8 + 3a3z - 8a3z3 + 9a3z5 - 3a3z7 - 2a4z2 + 3a4z4 - 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                 7 3     j = 6               9 5       j = 4             9 7         j = 2           9 9           j = 0         7 11             j = -2       4 7               j = -4     2 7                 j = -6   1 4                   j = -8   2                     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, 262]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 262]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 3, 13, 4], X[14, 6, 15, 5], X[22, 13, 9, 14], > X[16, 20, 17, 19], X[18, 8, 19, 7], X[6, 18, 7, 17], X[20, 16, 21, 15], > X[4, 22, 5, 21], X[2, 9, 3, 10], X[8, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 3, -7, 6, -11}, > {10, -1, 11, -2, 4, -3, 8, -5, 7, -6, 5, -8, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 6 11 14 3/2 5/2 q - ---- + ---- - ---- + ------- - 18 Sqrt[q] + 18 q - 16 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 12 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 2 4 -2 2 4 6 10 12 14 5 - q + q - -- + -- + q - q + 3 q - 2 q + 2 q - 3 q + 3 q - 8 6 q q 18 20 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 262]][a, z]
Out[8]=	3 3 5 5 1 a z z 3 z 3 z 3 3 3 z 2 z 5 -(---) + - - -- + - + a z - a z - -- + ---- + a z - a z + -- + ---- + a z a z z 3 a 5 a 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 262]][a, z]
Out[9]=	2 2 2 1 a z 2 z 4 z 3 2 z z 5 z 1 - --- - - + -- + --- + --- + 6 a z + 3 a z + 5 z - -- + -- + ---- - a z z 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 2 z 5 z 10 z 9 z 3 3 3 4 6 z > 2 a z + -- - ---- - ----- - ---- - 13 a z - 8 a z - 20 z + ---- - 7 5 3 a 6 a a a a 4 4 5 5 5 5 4 z 23 z 2 4 4 4 z 12 z 13 z 2 z 5 > ---- - ----- - 4 a z + 3 a z - -- + ----- + ----- + ---- + 11 a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 3 5 6 4 z 10 z 27 z 2 6 4 6 7 z 2 z > 9 a z + 23 z - ---- + ----- + ----- + 9 a z - a z - ---- - ---- + 6 4 2 5 3 a a a a a 7 8 8 9 9 9 z 7 3 7 8 7 z 11 z 2 8 4 z 7 z > ---- + a z - 3 a z - 8 z - ---- - ----- - 4 a z - ---- - ---- - a 4 2 3 a a a a 10 9 10 z > 3 a z - z - --- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 4 2 7 4 7 7 11 + 9 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + ---- + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t q t q t 2 4 4 2 6 2 6 3 8 3 8 4 > 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 3 q t + 10 4 10 5 12 5 14 6 > 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a262

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