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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-7/2 + 5q-5/2 - 12q-3/2 + 19q-1/2 - 26q1/2 + 29q3/2 - 29q5/2 + 24q7/2 - 18q9/2 + 10q11/2 - 4q13/2 + q15/2

A2 (sl(3)) Invariant:

q-10 - 3q-8 + 4q-6 - q-4 - 4q-2 + 7 - 4q2 + 5q4 + q6 + 5q10 - 5q12 + 4q14 - 3q18 + 3q20 - q24

HOMFLY-PT Polynomial:

a-7z - 2a-5z - 3a-5z3 - a-3z-1 + 2a-3z + 5a-3z3 + 3a-3z5 + a-1z-1 - a-1z - 5a-1z3 - 3a-1z5 - a-1z7 + az + az3 + az5

Kauffman Polynomial:

- a-8z2 + 2a-8z4 - a-8z6 + 2a-7z - 6a-7z3 + 8a-7z5 - 4a-7z7 + 4a-6z2 - 12a-6z4 + 16a-6z6 - 8a-6z8 + 3a-5z - 12a-5z3 + 10a-5z5 + 7a-5z7 - 8a-5z9 + 10a-4z2 - 41a-4z4 + 54a-4z6 - 19a-4z8 - 3a-4z10 + a-3z-1 - 15a-3z3 + 12a-3z5 + 23a-3z7 - 19a-3z9 - a-2 + 8a-2z2 - 41a-2z4 + 65a-2z6 - 27a-2z8 - 3a-2z10 + a-1z-1 - 15a-1z3 + 26a-1z5 - 11a-1z9 + 3z2 - 11z4 + 23z6 - 16z8 + az - 6az3 + 15az5 - 12az7 + 3a2z4 - 5a2z6 - a3z5

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 = 16                       1 j = 14                     3   j = 12                   7 1   j = 10                 11 3     j = 8               13 7       j = 6             16 11         j = 4           14 14           j = 2         12 15             j = 0       8 15               j = -2     4 11                 j = -4   1 8                   j = -6   4                     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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 180]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 180]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 4, 11, 3], X[14, 6, 15, 5], X[16, 8, 17, 7], > X[22, 18, 7, 17], X[12, 15, 13, 16], X[20, 10, 21, 9], X[18, 11, 19, 12], > X[6, 14, 1, 13], X[4, 20, 5, 19], X[2, 21, 3, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 2, -10, 3, -9}, > {4, -1, 7, -2, 8, -6, 9, -3, 6, -4, 5, -8, 10, -7, 11, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 5 12 19 3/2 5/2 7/2 -q + ---- - ---- + ------- - 26 Sqrt[q] + 29 q - 29 q + 24 q - 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 18 q + 10 q - 4 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 3 4 -4 4 2 4 6 10 12 14 7 + q - -- + -- - q - -- - 4 q + 5 q + q + 5 q - 5 q + 4 q - 8 6 2 q q q 18 20 24 > 3 q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 180]][a, z]
Out[8]=	3 3 3 5 1 1 z 2 z 2 z z 3 z 5 z 5 z 3 3 z -(----) + --- + -- - --- + --- - - + a z - ---- + ---- - ---- + a z + ---- - 3 a z 7 5 3 a 5 3 a 3 a z a a a a a a 5 7 3 z 5 z > ---- + a z - -- a a
In[9]:=	Kauffman[Link[11, Alternating, 180]][a, z]
Out[9]=	2 2 2 2 3 -2 1 1 2 z 3 z 2 z 4 z 10 z 8 z 6 z -a + ---- + --- + --- + --- + a z + 3 z - -- + ---- + ----- + ---- - ---- - 3 a z 7 5 8 6 4 2 7 a z a a a a a a a 3 3 3 4 4 4 4 12 z 15 z 15 z 3 4 2 z 12 z 41 z 41 z > ----- - ----- - ----- - 6 a z - 11 z + ---- - ----- - ----- - ----- + 5 3 a 8 6 4 2 a a a a a a 5 5 5 5 6 2 4 8 z 10 z 12 z 26 z 5 3 5 6 z > 3 a z + ---- + ----- + ----- + ----- + 15 a z - a z + 23 z - -- + 7 5 3 a 8 a a a a 6 6 6 7 7 7 16 z 54 z 65 z 2 6 4 z 7 z 23 z 7 8 > ----- + ----- + ----- - 5 a z - ---- + ---- + ----- - 12 a z - 16 z - 6 4 2 7 5 3 a a a a a a 8 8 8 9 9 9 10 10 8 z 19 z 27 z 8 z 19 z 11 z 3 z 3 z > ---- - ----- - ----- - ---- - ----- - ----- - ----- - ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 4 1 8 4 8 11 2 15 + 12 q + ----- + ----- + ----- + ----- + ----- + - + ---- + 15 q t + 8 4 6 3 4 3 4 2 2 2 t 2 q t q t q t q t q t q t 4 4 2 6 2 6 3 8 3 8 4 10 4 > 14 q t + 14 q t + 16 q t + 11 q t + 13 q t + 7 q t + 11 q t + 10 5 12 5 12 6 14 6 16 7 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a180

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