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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

- q-4 + q-3 - q-2 + q-1 + 1 + 2q2 - q3 + 2q4 - q5 + q6

A2 (sl(3)) Invariant:

- q-12 - q-10 - 2q-8 - q-6 - q-4 + q-2 + 3 + 3q2 + 5q4 + 4q6 + 5q8 + 4q10 + 3q12 + 3q14 + q16 + q18

HOMFLY-PT Polynomial:

2a-4z-2 + 5a-4 + 4a-4z2 + a-4z4 - 5a-2z-2 - 16a-2 - 17a-2z2 - 7a-2z4 - a-2z6 + 4z-2 + 15 + 16z2 + 7z4 + z6 - a2z-2 - 4a2 - 4a2z2 - a2z4

Kauffman Polynomial:

- 2a-6 + 6a-6z2 - 5a-6z4 + a-6z6 + a-5z + 2a-5z3 - 4a-5z5 + a-5z7 - 2a-4z-2 + 6a-4 - 7a-4z2 + 7a-4z4 - 5a-4z6 + a-4z8 + 5a-3z-1 - 16a-3z + 21a-3z3 - 12a-3z5 + 2a-3z7 - 5a-2z-2 + 20a-2 - 31a-2z2 + 22a-2z4 - 8a-2z6 + a-2z8 + 9a-1z-1 - 33a-1z + 37a-1z3 - 16a-1z5 + 2a-1z7 - 4z-2 + 17 - 25z2 + 20z4 - 8z6 + z8 + 5az-1 - 21az + 28az3 - 14az5 + 2az7 - a2z-2 + 4a2 - 7a2z2 + 10a2z4 - 6a2z6 + a2z8 + a3z-1 - 5a3z + 10a3z3 - 6a3z5 + a3z7

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 = 13                       1 j = 11                     1 1 j = 9                   1     j = 7               1 1 1     j = 5             1 3 1       j = 3           1   1         j = 1         1 4 2           j = -1       1   1             j = -3       1 1               j = -5   1 1                   j = -7                         j = -9 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 302]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 302]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[20, 13, 21, 14], X[22, 19, 11, 20], > X[15, 5, 16, 10], X[17, 9, 18, 8], X[7, 17, 8, 16], X[9, 19, 10, 18], > X[14, 21, 15, 22], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, > {-11, 2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-4 -3 -2 1 2 3 4 5 6 1 - q + q - q + - + 2 q - q + 2 q - q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 -6 -4 -2 2 4 6 8 10 3 - q - q - -- - q - q + q + 3 q + 5 q + 4 q + 5 q + 4 q + 8 q 12 14 16 18 > 3 q + 3 q + q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 302]][a, z]
Out[8]=	2 2 2 5 16 2 4 2 5 a 2 4 z 17 z 15 + -- - -- - 4 a + -- + ----- - ----- - -- + 16 z + ---- - ----- - 4 2 2 4 2 2 2 2 4 2 a a z a z a z z a a 4 4 6 2 2 4 z 7 z 2 4 6 z > 4 a z + 7 z + -- - ---- - a z + z - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 302]][a, z]
Out[9]=	2 3 2 6 20 2 4 2 5 a 5 9 5 a a 17 - -- + -- + -- + 4 a - -- - ----- - ----- - -- + ---- + --- + --- + -- + 6 4 2 2 4 2 2 2 2 3 a z z z a a a z a z a z z a z 2 2 2 z 16 z 33 z 3 2 6 z 7 z 31 z > -- - ---- - ---- - 21 a z - 5 a z - 25 z + ---- - ---- - ----- - 5 3 a 6 4 2 a a a a a 3 3 3 4 4 2 2 2 z 21 z 37 z 3 3 3 4 5 z 7 z > 7 a z + ---- + ----- + ----- + 28 a z + 10 a z + 20 z - ---- + ---- + 5 3 a 6 4 a a a a 4 5 5 5 6 22 z 2 4 4 z 12 z 16 z 5 3 5 6 z > ----- + 10 a z - ---- - ----- - ----- - 14 a z - 6 a z - 8 z + -- - 2 5 3 a 6 a a a a 6 6 7 7 7 8 8 5 z 8 z 2 6 z 2 z 2 z 7 3 7 8 z z > ---- - ---- - 6 a z + -- + ---- + ---- + 2 a z + a z + z + -- + -- + 4 2 5 3 a 4 2 a a a a a a 2 8 > a z
In[10]:=	Kh[L][q, t]
Out[10]=	1 3 1 1 1 1 1 1 q 5 - + 4 q + q + ----- + ----- + ----- + ----- + ---- + ---- + - + 2 q t + q t + q 9 5 5 4 5 3 3 2 2 3 t q t q t q t q t q t q t 3 2 5 2 7 2 5 3 7 3 7 4 9 4 11 5 11 6 > q t + 3 q t + q t + q t + q t + q t + q t + q t + q t + 13 6 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n302

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