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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X13,21,14,20 X19,11,20,22 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X21,15,22,14 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:

3q2 - 4q3 + 9q4 - 10q5 + 12q6 - 11q7 + 9q8 - 6q9 + 3q10 - q11

A2 (sl(3)) Invariant:

3q6 + q8 + 6q10 + 5q12 + 4q14 + 7q16 + q18 + 4q20 + q26 - 3q28 - q32 - q34

HOMFLY-PT Polynomial:

- a-10z-2 - 2a-10 - a-10z2 + 4a-8z-2 + 11a-8 + 10a-8z2 + 3a-8z4 - 5a-6z-2 - 18a-6 - 20a-6z2 - 10a-6z4 - 2a-6z6 + 2a-4z-2 + 9a-4 + 10a-4z2 + 3a-4z4

Kauffman Polynomial:

a-13z - 2a-13z3 + a-13z5 - a-12 + 3a-12z2 - 6a-12z4 + 3a-12z6 + a-11z-1 - 2a-11z + a-11z3 - 6a-11z5 + 4a-11z7 - a-10z-2 + 6a-10z2 - 9a-10z4 + 3a-10z8 + 5a-9z-1 - 19a-9z + 33a-9z3 - 26a-9z5 + 8a-9z7 + a-9z9 - 4a-8z-2 + 13a-8 - 19a-8z2 + 23a-8z4 - 17a-8z6 + 7a-8z8 + 9a-7z-1 - 35a-7z + 48a-7z3 - 28a-7z5 + 7a-7z7 + a-7z9 - 5a-6z-2 + 22a-6 - 38a-6z2 + 32a-6z4 - 14a-6z6 + 4a-6z8 + 5a-5z-1 - 19a-5z + 18a-5z3 - 9a-5z5 + 3a-5z7 - 2a-4z-2 + 11a-4 - 16a-4z2 + 6a-4z4

Khovanov Homology:

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   1 j = 21                 2   j = 19               4 1   j = 17             5 2     j = 15           7 5       j = 13         5 4         j = 11       5 7           j = 9     4 5             j = 7   1 6               j = 5 2 3                 j = 3 3

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, 301]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 301]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[13, 21, 14, 20], X[19, 11, 20, 22], > X[15, 5, 16, 10], X[17, 9, 18, 8], X[7, 17, 8, 16], X[9, 19, 10, 18], > X[21, 15, 22, 14], 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]=	2 3 4 5 6 7 8 9 10 11 3 q - 4 q + 9 q - 10 q + 12 q - 11 q + 9 q - 6 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	6 8 10 12 14 16 18 20 26 28 32 3 q + q + 6 q + 5 q + 4 q + 7 q + q + 4 q + q - 3 q - q - 34 > q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 301]][a, z]
Out[8]=	2 2 2 -2 11 18 9 1 4 5 2 z 10 z 20 z --- + -- - -- + -- - ------ + ----- - ----- + ----- - --- + ----- - ----- + 10 8 6 4 10 2 8 2 6 2 4 2 10 8 6 a a a a a z a z a z a z a a a 2 4 4 4 6 10 z 3 z 10 z 3 z 2 z > ----- + ---- - ----- + ---- - ---- 4 8 6 4 6 a a a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 301]][a, z]
Out[9]=	-12 13 22 11 1 4 5 2 1 5 9 -a + -- + -- + -- - ------ - ----- - ----- - ----- + ----- + ---- + ---- + 8 6 4 10 2 8 2 6 2 4 2 11 9 7 a a a a z a z a z a z a z a z a z 2 2 2 2 5 z 2 z 19 z 35 z 19 z 3 z 6 z 19 z 38 z > ---- + --- - --- - ---- - ---- - ---- + ---- + ---- - ----- - ----- - 5 13 11 9 7 5 12 10 8 6 a z a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 16 z 2 z z 33 z 48 z 18 z 6 z 9 z 23 z 32 z > ----- - ---- + --- + ----- + ----- + ----- - ---- - ---- + ----- + ----- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 7 6 z z 6 z 26 z 28 z 9 z 3 z 17 z 14 z 4 z > ---- + --- - ---- - ----- - ----- - ---- + ---- - ----- - ----- + ---- + 4 13 11 9 7 5 12 8 6 11 a a a a a a a a a a 7 7 7 8 8 8 9 9 8 z 7 z 3 z 3 z 7 z 4 z z z > ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 9 7 5 10 8 6 9 7 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 5 7 7 2 9 2 9 3 11 3 3 q + 2 q + 3 q t + q t + 6 q t + 4 q t + 5 q t + 5 q t + 11 4 13 4 13 5 15 5 15 6 17 6 > 7 q t + 5 q t + 4 q t + 7 q t + 5 q t + 5 q t + 17 7 19 7 19 8 21 8 23 9 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n301

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