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DrawMorseLink

PD Presentation:

X10,1,11,2 X12,4,13,3 X22,12,9,11 X14,6,15,5 X18,8,19,7 X20,18,21,17 X16,22,17,21 X2,9,3,10 X4,14,5,13 X6,16,7,15 X8,20,1,19

Gauss Code:

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

Jones Polynomial:

- q1/2 + 2q3/2 - 5q5/2 + 7q7/2 - 10q9/2 + 11q11/2 - 12q13/2 + 11q15/2 - 8q17/2 + 5q19/2 - 3q21/2 + q23/2

A2 (sl(3)) Invariant:

q2 + q6 + 2q8 + 3q12 - q14 + 2q16 + q18 - q20 + 2q22 - 2q24 + q26 + q32 - q34

HOMFLY-PT Polynomial:

a-9z + 3a-9z3 + a-9z5 - a-7z - 4a-7z3 - 4a-7z5 - a-7z7 - a-5z-1 - 2a-5z - 4a-5z3 - 4a-5z5 - a-5z7 + a-3z-1 + 4a-3z + 4a-3z3 + a-3z5

Kauffman Polynomial:

a-14z2 - a-14z4 - a-13z + 4a-13z3 - 3a-13z5 - a-12z2 + 4a-12z4 - 4a-12z6 - 2a-11z + 3a-11z3 + 2a-11z5 - 4a-11z7 - a-10z2 - 2a-10z4 + 5a-10z6 - 4a-10z8 - 2a-9z + 5a-9z3 - 5a-9z5 + 5a-9z7 - 3a-9z9 + 3a-8z2 - 11a-8z4 + 13a-8z6 - 3a-8z8 - a-8z10 - a-7z + 4a-7z3 - 12a-7z5 + 15a-7z7 - 5a-7z9 + 3a-6z2 - 12a-6z4 + 12a-6z6 - a-6z8 - a-6z10 - a-5z-1 + 5a-5z - 10a-5z3 + 3a-5z5 + 5a-5z7 - 2a-5z9 + a-4 + a-4z2 - 8a-4z4 + 8a-4z6 - 2a-4z8 - a-3z-1 + 5a-3z - 8a-3z3 + 5a-3z5 - a-3z7

Khovanov Homology:

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 24                       1 j = 22                     2   j = 20                   3 1   j = 18                 5 2     j = 16               6 3       j = 14             6 5         j = 12           6 7           j = 10         4 5             j = 8       3 6               j = 6     2 4                 j = 4   1 4                   j = 2   1                     j = 0 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, Alternating, 263]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 263]]
Out[4]=	PD[X[10, 1, 11, 2], X[12, 4, 13, 3], X[22, 12, 9, 11], X[14, 6, 15, 5], > X[18, 8, 19, 7], X[20, 18, 21, 17], X[16, 22, 17, 21], X[2, 9, 3, 10], > X[4, 14, 5, 13], X[6, 16, 7, 15], X[8, 20, 1, 19]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9, 4, -10, 5, -11}, > {8, -1, 3, -2, 9, -4, 10, -7, 6, -5, 11, -6, 7, -3}]
In[6]:=	Jones[L][q]
Out[6]=	3/2 5/2 7/2 9/2 11/2 13/2 -Sqrt[q] + 2 q - 5 q + 7 q - 10 q + 11 q - 12 q + 15/2 17/2 19/2 21/2 23/2 > 11 q - 8 q + 5 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	2 6 8 12 14 16 18 20 22 24 26 32 q + q + 2 q + 3 q - q + 2 q + q - q + 2 q - 2 q + q + q - 34 > q
In[8]:=	HOMFLYPT[Link[11, Alternating, 263]][a, z]
Out[8]=	3 3 3 3 5 5 1 1 z z 2 z 4 z 3 z 4 z 4 z 4 z z 4 z -(----) + ---- + -- - -- - --- + --- + ---- - ---- - ---- + ---- + -- - ---- - 5 3 9 7 5 3 9 7 5 3 9 7 a z a z a a a a a a a a a a 5 5 7 7 4 z z z z > ---- + -- - -- - -- 5 3 7 5 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 263]][a, z]
Out[9]=	2 2 2 2 -4 1 1 z 2 z 2 z z 5 z 5 z z z z 3 z a - ---- - ---- - --- - --- - --- - -- + --- + --- + --- - --- - --- + ---- + 5 3 13 11 9 7 5 3 14 12 10 8 a z a z a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 4 3 z z 4 z 3 z 5 z 4 z 10 z 8 z z 4 z 2 z > ---- + -- + ---- + ---- + ---- + ---- - ----- - ---- - --- + ---- - ---- - 6 4 13 11 9 7 5 3 14 12 10 a a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 11 z 12 z 8 z 3 z 2 z 5 z 12 z 3 z 5 z 4 z > ----- - ----- - ---- - ---- + ---- - ---- - ----- + ---- + ---- - ---- + 8 6 4 13 11 9 7 5 3 12 a a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 5 z 13 z 12 z 8 z 4 z 5 z 15 z 5 z z 4 z > ---- + ----- + ----- + ---- - ---- + ---- + ----- + ---- - -- - ---- - 10 8 6 4 11 9 7 5 3 10 a a a a a a a a a a 8 8 8 9 9 9 10 10 3 z z 2 z 3 z 5 z 2 z z z > ---- - -- - ---- - ---- - ---- - ---- - --- - --- 8 6 4 9 7 5 8 6 a a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 4 6 -2 q q 6 8 8 2 10 2 10 3 4 q + 2 q + t + -- + -- + 4 q t + 3 q t + 6 q t + 4 q t + 5 q t + t t 12 3 12 4 14 4 14 5 16 5 16 6 > 6 q t + 7 q t + 6 q t + 5 q t + 6 q t + 3 q t + 18 6 18 7 20 7 20 8 22 8 24 9 > 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a263

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