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DrawMorseLink

PD Presentation:

X8192 X10,4,11,3 X22,10,7,9 X2738 X15,5,16,4 X5,13,6,12 X16,12,17,11 X6,18,1,17 X19,14,20,15 X13,20,14,21 X21,19,22,18

Gauss Code:

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

Jones Polynomial:

- q-5/2 + 2q-3/2 - 4q-1/2 + 5q1/2 - 6q3/2 + 6q5/2 - 6q7/2 + 4q9/2 - 3q11/2 + q13/2

A2 (sl(3)) Invariant:

q-8 + q-6 + q-2 + q2 + 2q8 - q10 + 2q12 + q16 + q18 + q22 - q24

HOMFLY-PT Polynomial:

a-7z - a-5z-1 - 3a-5z - 4a-5z3 - a-5z5 + 2a-3z-1 + 7a-3z + 8a-3z3 + 5a-3z5 + a-3z7 - 2a-1z-1 - 7a-1z - 8a-1z3 - 2a-1z5 + az-1 + 3az + az3

Kauffman Polynomial:

- a-8z2 + 2a-7z - 3a-7z3 - a-6z6 - a-5z-1 + 7a-5z - 13a-5z3 + 10a-5z5 - 3a-5z7 + 3a-4z2 - 9a-4z4 + 10a-4z6 - 3a-4z8 - 2a-3z-1 + 13a-3z - 27a-3z3 + 20a-3z5 - 2a-3z7 - a-3z9 - a-2 + 6a-2z2 - 20a-2z4 + 20a-2z6 - 5a-2z8 - 2a-1z-1 + 13a-1z - 25a-1z3 + 15a-1z5 - a-1z9 + 4z2 - 11z4 + 9z6 - 2z8 - az-1 + 5az - 8az3 + 5az5 - az7

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 14                   1 j = 12                 2   j = 10               2 1   j = 8             4 2     j = 6           3 3       j = 4         3 3         j = 2       3 4           j = 0     1 2             j = -2   1 3               j = -4   1                 j = -6 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 128]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 128]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[22, 10, 7, 9], X[2, 7, 3, 8], > X[15, 5, 16, 4], X[5, 13, 6, 12], X[16, 12, 17, 11], X[6, 18, 1, 17], > X[19, 14, 20, 15], X[13, 20, 14, 21], X[21, 19, 22, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 2, 5, -6, -8}, > {4, -1, 3, -2, 7, 6, -10, 9, -5, -7, 8, 11, -9, 10, -11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(5/2) 2 4 3/2 5/2 7/2 9/2 -q + ---- - ------- + 5 Sqrt[q] - 6 q + 6 q - 6 q + 4 q - 3/2 Sqrt[q] q 11/2 13/2 > 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-8 -6 -2 2 8 10 12 16 18 22 24 q + q + q + q + 2 q - q + 2 q + q + q + q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 128]][a, z]
Out[8]=	3 3 3 1 2 2 a z 3 z 7 z 7 z 4 z 8 z 8 z -(----) + ---- - --- + - + -- - --- + --- - --- + 3 a z - ---- + ---- - ---- + 5 3 a z z 7 5 3 a 5 3 a a z a z a a a a a 5 5 5 7 3 z 5 z 2 z z > a z - -- + ---- - ---- + -- 5 3 a 3 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 128]][a, z]
Out[9]=	2 -2 1 2 2 a 2 z 7 z 13 z 13 z 2 z -a - ---- - ---- - --- - - + --- + --- + ---- + ---- + 5 a z + 4 z - -- + 5 3 a z z 7 5 3 a 8 a z a z a a a a 2 2 3 3 3 3 4 3 z 6 z 3 z 13 z 27 z 25 z 3 4 9 z > ---- + ---- - ---- - ----- - ----- - ----- - 8 a z - 11 z - ---- - 4 2 7 5 3 a 4 a a a a a a 4 5 5 5 6 6 6 7 20 z 10 z 20 z 15 z 5 6 z 10 z 20 z 3 z > ----- + ----- + ----- + ----- + 5 a z + 9 z - -- + ----- + ----- - ---- - 2 5 3 a 6 4 2 5 a a a a a a a 7 8 8 9 9 2 z 7 8 3 z 5 z z z > ---- - a z - 2 z - ---- - ---- - -- - -- 3 4 2 3 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 -2 3 2 3 q 4 4 q + 3 q + ----- + ----- + ----- + t + ----- + - + ---- + 3 q t + 6 4 4 3 2 3 2 2 t t q t q t q t q t 6 6 2 8 2 8 3 10 3 10 4 12 4 14 5 > 3 q t + 3 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n128

L11n127 L11n129