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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 3q-3/2 - 8q-1/2 + 12q1/2 - 18q3/2 + 20q5/2 - 21q7/2 + 19q9/2 - 14q11/2 + 9q13/2 - 4q15/2 + q17/2

A2 (sl(3)) Invariant:

q-8 + 4q-2 - 1 + 4q2 + 4q4 - q6 + 5q8 - 4q10 + q12 - 2q14 - 3q16 + 3q18 - 3q20 + q22 + q24 - q26

HOMFLY-PT Polynomial:

a-7z + a-7z3 + a-5z-1 - a-5z - 4a-5z3 - 2a-5z5 - 2a-3z-1 + 3a-3z3 + 3a-3z5 + a-3z7 - 3a-1z - 5a-1z3 - 2a-1z5 + az-1 + 2az + az3

Kauffman Polynomial:

- a-10z4 + a-9z3 - 4a-9z5 - 3a-8z2 + 8a-8z4 - 9a-8z6 + a-7z - 8a-7z3 + 17a-7z5 - 13a-7z7 - 2a-6 + 4a-6z2 - 8a-6z4 + 19a-6z6 - 13a-6z8 + a-5z-1 - 8a-5z3 + 15a-5z5 + 4a-5z7 - 8a-5z9 - 5a-4 + 18a-4z2 - 42a-4z4 + 51a-4z6 - 16a-4z8 - 2a-4z10 + 2a-3z-1 - 2a-3z - 3a-3z3 - 9a-3z5 + 27a-3z7 - 12a-3z9 - 3a-2 + 13a-2z2 - 35a-2z4 + 33a-2z6 - 6a-2z8 - 2a-2z10 + 3a-1z - 10a-1z3 + a-1z5 + 9a-1z7 - 4a-1z9 + 1 + 2z2 - 10z4 + 10z6 - 3z8 - az-1 + 4az - 6az3 + 4az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 18                       1 j = 16                     3   j = 14                   6 1   j = 12                 8 3     j = 10               11 6       j = 8             10 8         j = 6           10 11           j = 4         8 10             j = 2       5 11               j = 0     3 7                 j = -2   1 6                   j = -4   2                     j = -6 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, 213]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 213]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[18, 10, 19, 9], X[22, 20, 7, 19], > X[20, 14, 21, 13], X[14, 22, 15, 21], X[10, 16, 11, 15], X[16, 6, 17, 5], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 18, 1, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, > {9, -1, 3, -7, 10, -2, 5, -6, 7, -8, 11, -3, 4, -5, 6, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 3 8 3/2 5/2 7/2 -q + ---- - ------- + 12 Sqrt[q] - 18 q + 20 q - 21 q + 3/2 Sqrt[q] q 9/2 11/2 13/2 15/2 17/2 > 19 q - 14 q + 9 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 4 2 4 6 8 10 12 14 16 18 -1 + q + -- + 4 q + 4 q - q + 5 q - 4 q + q - 2 q - 3 q + 3 q - 2 q 20 22 24 26 > 3 q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 213]][a, z]
Out[8]=	3 3 3 3 1 2 a z z 3 z z 4 z 3 z 5 z 3 ---- - ---- + - + -- - -- - --- + 2 a z + -- - ---- + ---- - ---- + a z - 5 3 z 7 5 a 7 5 3 a a z a z a a a a a 5 5 5 7 2 z 3 z 2 z z > ---- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 213]][a, z]
Out[9]=	2 2 5 3 1 2 a z 2 z 3 z 2 3 z 1 - -- - -- - -- + ---- + ---- - - + -- - --- + --- + 4 a z + 2 z - ---- + 6 4 2 5 3 z 7 3 a 8 a a a a z a z a a a 2 2 2 3 3 3 3 3 4 z 18 z 13 z z 8 z 8 z 3 z 10 z 3 4 > ---- + ----- + ----- + -- - ---- - ---- - ---- - ----- - 6 a z - 10 z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 4 5 5 5 5 5 z 8 z 8 z 42 z 35 z 4 z 17 z 15 z 9 z z > --- + ---- - ---- - ----- - ----- - ---- + ----- + ----- - ---- + -- + 10 8 6 4 2 9 7 5 3 a a a a a a a a a a 6 6 6 6 7 7 7 5 6 9 z 19 z 51 z 33 z 13 z 4 z 27 z > 4 a z + 10 z - ---- + ----- + ----- + ----- - ----- + ---- + ----- + 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 9 z 7 8 13 z 16 z 6 z 8 z 12 z 4 z 2 z > ---- - a z - 3 z - ----- - ----- - ---- - ---- - ----- - ---- - ----- - a 6 4 2 5 3 a 4 a a a a a a 10 2 z > ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 2 1 3 6 7 5 q 4 11 q + 8 q + ----- + ----- + ----- + -- + ----- + - + ---- + 10 q t + 6 4 4 3 2 3 2 2 2 t t q t q t q t t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 > 10 q t + 11 q t + 10 q t + 8 q t + 11 q t + 6 q t + 8 q t + 12 5 14 5 14 6 16 6 18 7 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a213

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