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DrawMorseLink

PD Presentation:

X8192 X16,8,17,7 X10,4,11,3 X2,18,3,17 X18,9,19,10 X20,12,21,11 X14,6,15,5 X22,15,13,16 X6,14,1,13 X4,19,5,20 X12,22,7,21

Gauss Code:

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

Jones Polynomial:

q-3 - 5q-2 + 11q-1 - 17 + 24q - 26q2 + 27q3 - 22q4 + 17q5 - 9q6 + 4q7 - q8

A2 (sl(3)) Invariant:

q-8 - 3q-6 + 3q-4 - 2q-2 + 5q2 - 4q4 + 8q6 + 6q10 + 6q12 + 7q16 - q18 + q20 + q22 - q24

HOMFLY-PT Polynomial:

a-6z-2 - 2a-6z2 - a-6z4 - 2a-4z-2 - 2a-4 + 4a-4z2 + 6a-4z4 + 2a-4z6 + a-2z-2 + 2a-2 - a-2z2 - 5a-2z4 - 4a-2z6 - a-2z8 + 2z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 5a-8z4 + 4a-8z6 + 4a-7z3 - 10a-7z5 + 8a-7z7 + a-6z-2 - 2a-6 - 3a-6z2 + 11a-6z4 - 16a-6z6 + 11a-6z8 - 2a-5z-1 + 2a-5z + 5a-5z3 - 6a-5z5 - 6a-5z7 + 9a-5z9 + 2a-4z-2 - 3a-4 - 14a-4z2 + 48a-4z4 - 54a-4z6 + 19a-4z8 + 3a-4z10 - 2a-3z-1 + 2a-3z + 5a-3z3 + 6a-3z5 - 30a-3z7 + 18a-3z9 + a-2z-2 - 2a-2 - 11a-2z2 + 46a-2z4 - 57a-2z6 + 18a-2z8 + 3a-2z10 + 8a-1z3 - 8a-1z5 - 11a-1z7 + 9a-1z9 - 2z2 + 13z4 - 22z6 + 10z8 + 3az3 - 9az5 + 5az7 - a2z4 + a2z6

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 = 17                       1 j = 15                     3   j = 13                   6 1   j = 11                 11 3     j = 9               12 7       j = 7             15 10         j = 5           13 14           j = 3         11 13             j = 1       7 14               j = -1     4 10                 j = -3   1 7                   j = -5   4                     j = -7 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, Alternating, 505]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 505]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 8, 17, 7], X[10, 4, 11, 3], X[2, 18, 3, 17], > X[18, 9, 19, 10], X[20, 12, 21, 11], X[14, 6, 15, 5], X[22, 15, 13, 16], > X[6, 14, 1, 13], X[4, 19, 5, 20], X[12, 22, 7, 21]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10, 7, -9}, {2, -1, 5, -3, 6, -11}, > {9, -7, 8, -2, 4, -5, 10, -6, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-3 5 11 2 3 4 5 6 7 8 -17 + q - -- + -- + 24 q - 26 q + 27 q - 22 q + 17 q - 9 q + 4 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 3 3 2 2 4 6 10 12 16 18 20 q - -- + -- - -- + 5 q - 4 q + 8 q + 6 q + 6 q + 7 q - q + q + 6 4 2 q q q 22 24 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 505]][a, z]
Out[8]=	2 2 2 4 4 4 -2 2 1 2 1 2 z 4 z z 4 z 6 z 5 z -- + -- + ----- - ----- + ----- - ---- + ---- - -- + 2 z - -- + ---- - ---- + 4 2 6 2 4 2 2 2 6 4 2 6 4 2 a a a z a z a z a a a a a a 6 6 8 6 2 z 4 z z > z + ---- - ---- - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, Alternating, 505]][a, z]
Out[9]=	2 -2 3 2 1 2 1 2 2 2 z 2 z 2 2 z -- - -- - -- + ----- + ----- + ----- - ---- - ---- + --- + --- - 2 z + ---- - 6 4 2 6 2 4 2 2 2 5 3 5 3 8 a a a a z a z a z a z a z a a a 2 2 2 3 3 3 3 3 3 z 14 z 11 z z 4 z 5 z 5 z 8 z 3 4 > ---- - ----- - ----- - -- + ---- + ---- + ---- + ---- + 3 a z + 13 z - 6 4 2 9 7 5 3 a a a a a a a a 4 4 4 4 5 5 5 5 5 5 z 11 z 48 z 46 z 2 4 z 10 z 6 z 6 z 8 z > ---- + ----- + ----- + ----- - a z + -- - ----- - ---- + ---- - ---- - 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 5 6 4 z 16 z 54 z 57 z 2 6 8 z 6 z > 9 a z - 22 z + ---- - ----- - ----- - ----- + a z + ---- - ---- - 8 6 4 2 7 5 a a a a a a 7 7 8 8 8 9 9 30 z 11 z 7 8 11 z 19 z 18 z 9 z 18 z > ----- - ----- + 5 a z + 10 z + ----- + ----- + ----- + ---- + ----- + 3 a 6 4 2 5 3 a a a a a a 9 10 10 9 z 3 z 3 z > ---- + ----- + ----- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 4 1 7 4 10 7 q 3 14 q + 11 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 13 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 13 q t + 14 q t + 15 q t + 10 q t + 12 q t + 7 q t + 11 q t + 11 5 13 5 13 6 15 6 17 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 L11a505

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