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DrawMorseLink

PD Presentation:

X6172 X14,4,15,3 X16,8,17,7 X22,14,5,13 X8,22,9,21 X20,10,21,9 X18,12,19,11 X12,18,13,17 X10,20,11,19 X2536 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

- q-1/2 + 2q1/2 - 5q3/2 + 6q5/2 - 9q7/2 + 10q9/2 - 10q11/2 + 8q13/2 - 6q15/2 + 4q17/2 - 2q19/2 + q21/2

A2 (sl(3)) Invariant:

q-2 + q2 + 3q4 + 3q8 + q10 + 2q14 + 2q18 - q22 + q24 - 2q26 - q28 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 3a-12z2 + 4a-12z4 - a-12z6 - 4a-11z3 + 7a-11z5 - 2a-11z7 - 3a-10 + 11a-10z2 - 15a-10z4 + 12a-10z6 - 3a-10z8 + a-9z-1 - 5a-9z + 17a-9z3 - 21a-9z5 + 13a-9z7 - 3a-9z9 - 7a-8 + 29a-8z2 - 39a-8z4 + 18a-8z6 - a-8z8 - a-8z10 + 2a-7z-1 - 10a-7z + 25a-7z3 - 38a-7z5 + 22a-7z7 - 5a-7z9 - 4a-6 + 15a-6z2 - 23a-6z4 + 9a-6z6 - a-6z10 + a-5z-1 - 5a-5z + 7a-5z3 - 8a-5z5 + 5a-5z7 - 2a-5z9 + a-4z4 + 2a-4z6 - 2a-4z8 + a-3z-1 - 3a-3z + 6a-3z3 + a-3z5 - 2a-3z7 - a-2 + 4a-2z4 - 2a-2z6 + a-1z-1 - 3a-1z + 3a-1z3 - a-1z5

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 = 22                       1 j = 20                     1   j = 18                   3 1   j = 16                 3 1     j = 14               5 3       j = 12             5 3         j = 10           5 5           j = 8         4 5             j = 6       2 5               j = 4     3 4                 j = 2   1 4                   j = 0   1                     j = -2 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, 117]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 117]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[16, 8, 17, 7], X[22, 14, 5, 13], > X[8, 22, 9, 21], X[20, 10, 21, 9], X[18, 12, 19, 11], X[12, 18, 13, 17], > X[10, 20, 11, 19], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -5, 6, -9, 7, -8, 4, -2, 11, -3, 8, -7, > 9, -6, 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	1 3/2 5/2 7/2 9/2 11/2 -(-------) + 2 Sqrt[q] - 5 q + 6 q - 9 q + 10 q - 10 q + Sqrt[q] 13/2 15/2 17/2 19/2 21/2 > 8 q - 6 q + 4 q - 2 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-2 2 4 8 10 14 18 22 24 26 28 32 q + q + 3 q + 3 q + q + 2 q + 2 q - q + q - 2 q - q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 117]][a, z]
Out[8]=	3 3 3 1 2 1 1 1 2 z 4 z z z 2 z z 3 z z ---- - ---- + ---- - ---- + --- + --- - --- + -- - -- + --- + -- - ---- - -- - 9 7 5 3 a z 9 7 5 3 a 9 7 5 a z a z a z a z a a a a a a a 3 3 5 5 5 2 z z z z z > ---- + -- - -- - -- - -- 3 a 7 5 3 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 117]][a, z]
Out[9]=	-3 7 4 -2 1 2 1 1 1 5 z 10 z 5 z --- - -- - -- - a + ---- + ---- + ---- + ---- + --- - --- - ---- - --- - 10 8 6 9 7 5 3 a z 9 7 5 a a a a z a z a z a z a a a 2 2 2 2 3 3 3 3 3 z 3 z 3 z 11 z 29 z 15 z 4 z 17 z 25 z 7 z > --- - --- - ---- + ----- + ----- + ----- - ---- + ----- + ----- + ---- + 3 a 12 10 8 6 11 9 7 5 a a a a a a a a a 3 3 4 4 4 4 4 4 5 5 6 z 3 z 4 z 15 z 39 z 23 z z 4 z 7 z 21 z > ---- + ---- + ---- - ----- - ----- - ----- + -- + ---- + ---- - ----- - 3 a 12 10 8 6 4 2 11 9 a a a a a a a a a 5 5 5 5 6 6 6 6 6 6 7 38 z 8 z z z z 12 z 18 z 9 z 2 z 2 z 2 z > ----- - ---- + -- - -- - --- + ----- + ----- + ---- + ---- - ---- - ---- + 7 5 3 a 12 10 8 6 4 2 11 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 9 10 13 z 22 z 5 z 2 z 3 z z 2 z 3 z 5 z 2 z z > ----- + ----- + ---- - ---- - ---- - -- - ---- - ---- - ---- - ---- - --- - 9 7 5 3 10 8 4 9 7 5 8 a a a a a a a a a a a 10 z > --- 6 a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 q 4 6 6 2 8 2 8 3 4 q + 3 q + ----- + - + -- + 4 q t + 2 q t + 5 q t + 4 q t + 5 q t + 2 2 t t q t 10 3 10 4 12 4 12 5 14 5 14 6 > 5 q t + 5 q t + 5 q t + 3 q t + 5 q t + 3 q t + 16 6 16 7 18 7 18 8 20 8 22 9 > 3 q t + q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a117

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