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DrawMorseLink

PD Presentation:

X6172 X10,4,11,3 X12,8,13,7 X15,17,16,22 X9,18,10,19 X17,8,18,9 X13,21,14,20 X21,15,22,14 X19,5,20,16 X2536 X4,12,1,11

Gauss Code:

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

Jones Polynomial:

q-1 - 2 + 6q - 6q2 + 9q3 - 8q4 + 7q5 - 5q6 + 3q7 - q8

A2 (sl(3)) Invariant:

q-4 + q-2 + 1 + 5q2 + 3q4 + 6q6 + 5q8 + 2q10 + 3q12 - q14 + 2q16 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + a-4z-2 + 3a-4 + 6a-4z2 + 4a-4z4 + a-4z6 - 2a-2z-2 - 4a-2 - 5a-2z2 - 2a-2z4 + z-2 + 2 + z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 2a-8z2 - 7a-8z4 + 3a-8z6 - a-7z + 6a-7z3 - 10a-7z5 + 4a-7z7 + 2a-6 - 5a-6z2 + 7a-6z4 - 7a-6z6 + 3a-6z8 - 3a-5z + 9a-5z3 - 6a-5z5 + a-5z7 + a-5z9 - a-4z-2 + 7a-4 - 21a-4z2 + 27a-4z4 - 14a-4z6 + 4a-4z8 + 2a-3z-1 - 4a-3z - a-3z3 + 7a-3z5 - 3a-3z7 + a-3z9 - 2a-2z-2 + 7a-2 - 17a-2z2 + 14a-2z4 - 4a-2z6 + a-2z8 + 2a-1z-1 - 2a-1z - 2a-1z3 + 2a-1z5 - z-2 + 3 - 3z2 + z4

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                   1 j = 15                 2   j = 13               3 1   j = 11             4 2     j = 9           4 3       j = 7         5 4         j = 5       3 6           j = 3     3 3             j = 1   1 5               j = -1   1                 j = -3 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, 364]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 364]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 8, 13, 7], X[15, 17, 16, 22], > X[9, 18, 10, 19], X[17, 8, 18, 9], X[13, 21, 14, 20], X[21, 15, 22, 14], > X[19, 5, 20, 16], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {-6, 5, -9, 7, -8, 4}, > {10, -1, 3, 6, -5, -2, 11, -3, -7, 8, -4, 9}]
In[6]:=	Jones[L][q]
Out[6]=	1 2 3 4 5 6 7 8 -2 + - + 6 q - 6 q + 9 q - 8 q + 7 q - 5 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 2 4 6 8 10 12 14 16 20 1 + q + q + 5 q + 3 q + 6 q + 5 q + 2 q + 3 q - q + 2 q - q + 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 364]][a, z]
Out[8]=	2 2 2 4 4 -6 3 4 -2 1 2 2 2 z 6 z 5 z z 4 z 2 - a + -- - -- + z + ----- - ----- + z - ---- + ---- - ---- - -- + ---- - 4 2 4 2 2 2 6 4 2 6 4 a a a z a z a a a a a 4 6 2 z z > ---- + -- 2 4 a a
In[9]:=	Kauffman[Link[11, NonAlternating, 364]][a, z]
Out[9]=	2 7 7 -2 1 2 2 2 z 3 z 4 z 2 z 3 + -- + -- + -- - z - ----- - ----- + ---- + --- - -- - --- - --- - --- - 6 4 2 4 2 2 2 3 a z 7 5 3 a a a a a z a z a z a a a 2 2 2 2 3 3 3 3 3 2 2 z 5 z 21 z 17 z 2 z 6 z 9 z z 2 z 4 > 3 z + ---- - ---- - ----- - ----- - ---- + ---- + ---- - -- - ---- + z - 8 6 4 2 9 7 5 3 a a a a a a a a a 4 4 4 4 5 5 5 5 5 6 7 z 7 z 27 z 14 z z 10 z 6 z 7 z 2 z 3 z > ---- + ---- + ----- + ----- + -- - ----- - ---- + ---- + ---- + ---- - 8 6 4 2 9 7 5 3 a 8 a a a a a a a a a 6 6 6 7 7 7 8 8 8 9 9 7 z 14 z 4 z 4 z z 3 z 3 z 4 z z z z > ---- - ----- - ---- + ---- + -- - ---- + ---- + ---- + -- + -- + -- 6 4 2 7 5 3 6 4 2 5 3 a a a a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 q 3 5 5 2 7 2 7 3 5 q + 3 q + ----- + --- + - + 3 q t + 3 q t + 6 q t + 5 q t + 4 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 17 7 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n364

L11n363 L11n365