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DrawMorseLink

PD Presentation:

X6172 X14,5,15,6 X12,4,13,3 X2,9,3,10 X7,19,8,18 X17,9,18,8 X10,13,5,14 X19,22,20,17 X21,11,22,16 X11,21,12,20 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

q-7/2 - 4q-5/2 + 5q-3/2 - 8q-1/2 + 7q1/2 - 9q3/2 + 5q5/2 - 6q7/2 + 2q9/2 - q11/2

A2 (sl(3)) Invariant:

- q-12 + 3q-8 + q-6 + 4q-4 + 4q-2 + 5 + 10q2 + 10q4 + 14q6 + 11q8 + 8q10 + 7q12 + 2q14 + 2q16 + q18

HOMFLY-PT Polynomial:

- a-5z-3 - 2a-5z-1 - a-5z + 3a-3z-3 + 7a-3z-1 + 8a-3z + 3a-3z3 - 3a-1z-3 - 8a-1z-1 - 12a-1z - 8a-1z3 - 2a-1z5 + az-3 + 3az-1 + 6az + 3az3 - a3z

Kauffman Polynomial:

a-5z-3 - 5a-5z-1 + 10a-5z - 10a-5z3 + 5a-5z5 - a-5z7 - 3a-4z-2 + 10a-4 - 9a-4z2 - 3a-4z4 + 7a-4z6 - 2a-4z8 + 3a-3z-3 - 12a-3z-1 + 27a-3z - 42a-3z3 + 27a-3z5 - 3a-3z7 - a-3z9 - 6a-2z-2 + 19a-2 - 18a-2z2 - 6a-2z4 + 19a-2z6 - 6a-2z8 + 3a-1z-3 - 12a-1z-1 + 31a-1z - 52a-1z3 + 39a-1z5 - 7a-1z7 - a-1z9 - 3z-2 + 10 - 10z2 - z4 + 10z6 - 4z8 + az-3 - 5az-1 + 16az - 24az3 + 17az5 - 5az7 - 2a2z2 + 2a2z4 - 2a2z6 + 2a3z - 4a3z3 - a4z2

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 12                   1 j = 10                 1   j = 8               5 1   j = 6             2 3     j = 4           7 3       j = 2       1 3 4         j = 0       6 5           j = -2   1 2 4             j = -4   3 4               j = -6   3                 j = -8 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 455]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 455]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 5, 15, 6], X[12, 4, 13, 3], X[2, 9, 3, 10], > X[7, 19, 8, 18], X[17, 9, 18, 8], X[10, 13, 5, 14], X[19, 22, 20, 17], > X[21, 11, 22, 16], X[11, 21, 12, 20], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -11}, {2, -1, -5, 6, 4, -7}, {-10, -3, 7, -2, 11, 9}, > {-6, 5, -8, 10, -9, 8}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 4 5 8 3/2 5/2 7/2 q - ---- + ---- - ------- + 7 Sqrt[q] - 9 q + 5 q - 6 q + 5/2 3/2 Sqrt[q] q q 9/2 11/2 > 2 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 3 -6 4 4 2 4 6 8 10 12 5 - q + -- + q + -- + -- + 10 q + 10 q + 14 q + 11 q + 8 q + 7 q + 8 4 2 q q q 14 16 18 > 2 q + 2 q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 455]][a, z]
Out[8]=	1 3 3 a 2 7 8 3 a z 8 z 12 z -(-----) + ----- - ---- + -- - ---- + ---- - --- + --- - -- + --- - ---- + 5 3 3 3 3 3 5 3 a z z 5 3 a a z a z a z z a z a z a a 3 3 5 3 3 z 8 z 3 2 z > 6 a z - a z + ---- - ---- + 3 a z - ---- 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 455]][a, z]
Out[9]=	10 19 1 3 3 a 3 3 6 5 12 10 + -- + -- + ----- + ----- + ---- + -- - -- - ----- - ----- - ---- - ---- - 4 2 5 3 3 3 3 3 2 4 2 2 2 5 3 a a a z a z a z z z a z a z a z a z 2 2 12 5 a 10 z 27 z 31 z 3 2 9 z 18 z > --- - --- + ---- + ---- + ---- + 16 a z + 2 a z - 10 z - ---- - ----- - a z z 5 3 a 4 2 a a a a 3 3 3 4 2 2 4 2 10 z 42 z 52 z 3 3 3 4 3 z > 2 a z - a z - ----- - ----- - ----- - 24 a z - 4 a z - z - ---- - 5 3 a 4 a a a 4 5 5 5 6 6 6 z 2 4 5 z 27 z 39 z 5 6 7 z 19 z > ---- + 2 a z + ---- + ----- + ----- + 17 a z + 10 z + ---- + ----- - 2 5 3 a 4 2 a a a a a 7 7 7 8 8 9 9 2 6 z 3 z 7 z 7 8 2 z 6 z z z > 2 a z - -- - ---- - ---- - 5 a z - 4 z - ---- - ---- - -- - -- 5 3 a 4 2 3 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	4 2 1 3 3 1 4 2 2 6 + -- + q + ----- + ----- + ----- + ----- + ---- + ---- + 5 t + 3 q t + 2 8 3 6 2 4 2 2 2 4 2 q q t q t q t q t q t q t 2 2 4 2 4 3 6 3 6 4 8 4 8 5 > 4 q t + 7 q t + 3 q t + 2 q t + 3 q t + 5 q t + q t + 10 5 12 6 > q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n455

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