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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9/2 - 3q-7/2 + 7q-5/2 - 13q-3/2 + 17q-1/2 - 21q1/2 + 20q3/2 - 19q5/2 + 14q7/2 - 8q9/2 + 4q11/2 - q13/2

A2 (sl(3)) Invariant:

- q-14 - 3q-8 + 4q-6 + q-2 + 6 - q2 + 6q4 - q6 + q8 + q10 - 5q12 + 2q14 - q16 - q18 + q20

HOMFLY-PT Polynomial:

- a-5z - a-5z3 + a-3z-1 + 5a-3z + 5a-3z3 + 2a-3z5 - 4a-1z-1 - 10a-1z - 9a-1z3 - 4a-1z5 - a-1z7 + 4az-1 + 8az + 6az3 + 2az5 - a3z-1 - 2a3z - a3z3

Kauffman Polynomial:

a-7z3 - a-7z5 - 3a-6z2 + 6a-6z4 - 4a-6z6 + 2a-5z - 6a-5z3 + 10a-5z5 - 7a-5z7 + a-4 - 9a-4z2 + 10a-4z4 + 3a-4z6 - 7a-4z8 - a-3z-1 + 10a-3z - 26a-3z3 + 31a-3z5 - 10a-3z7 - 4a-3z9 + 4a-2 - 13a-2z2 + 3a-2z4 + 21a-2z6 - 14a-2z8 - a-2z10 - 4a-1z-1 + 19a-1z - 40a-1z3 + 38a-1z5 - 5a-1z7 - 7a-1z9 + 7 - 14z2 + z4 + 20z6 - 11z8 - z10 - 4az-1 + 15az - 29az3 + 26az5 - 5az7 - 3az9 + 4a2 - 10a2z2 + 5a2z4 + 5a2z6 - 4a2z8 - a3z-1 + 4a3z - 8a3z3 + 8a3z5 - 3a3z7 + a4 - 3a4z2 + 3a4z4 - a4z6

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 14                       1 j = 12                     3   j = 10                   5 1   j = 8                 9 3     j = 6               10 5       j = 4             10 9         j = 2           11 10           j = 0         8 12             j = -2       5 9               j = -4     2 8                 j = -6   1 5                   j = -8   2                     j = -10 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, 40]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 40]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 10, 13, 9], X[18, 15, 19, 16], > X[16, 7, 17, 8], X[8, 17, 9, 18], X[22, 20, 5, 19], X[20, 14, 21, 13], > X[14, 22, 15, 21], X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -6, 3, -2, 11, -3, 8, -9, 4, -5, 6, -4, > 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(9/2) 3 7 13 17 3/2 5/2 q - ---- + ---- - ---- + ------- - 21 Sqrt[q] + 20 q - 19 q + 7/2 5/2 3/2 Sqrt[q] q q q 7/2 9/2 11/2 13/2 > 14 q - 8 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 3 4 -2 2 4 6 8 10 12 14 16 6 - q - -- + -- + q - q + 6 q - q + q + q - 5 q + 2 q - q - 8 6 q q 18 20 > q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 40]][a, z]
Out[8]=	3 3 3 3 1 4 4 a a z 5 z 10 z 3 z 5 z 9 z ---- - --- + --- - -- - -- + --- - ---- + 8 a z - 2 a z - -- + ---- - ---- + 3 a z z z 5 3 a 5 3 a a z a a a a 5 5 7 3 3 3 2 z 4 z 5 z > 6 a z - a z + ---- - ---- + 2 a z - -- 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 40]][a, z]
Out[9]=	3 -4 4 2 4 1 4 4 a a 2 z 10 z 19 z 7 + a + -- + 4 a + a - ---- - --- - --- - -- + --- + ---- + ---- + 15 a z + 2 3 a z z z 5 3 a a a z a a 2 2 2 3 3 3 2 3 z 9 z 13 z 2 2 4 2 z 6 z > 4 a z - 14 z - ---- - ---- - ----- - 10 a z - 3 a z + -- - ---- - 6 4 2 7 5 a a a a a 3 3 4 4 4 26 z 40 z 3 3 3 4 6 z 10 z 3 z 2 4 > ----- - ----- - 29 a z - 8 a z + z + ---- + ----- + ---- + 5 a z + 3 a 6 4 2 a a a a 5 5 5 5 6 4 4 z 10 z 31 z 38 z 5 3 5 6 4 z > 3 a z - -- + ----- + ----- + ----- + 26 a z + 8 a z + 20 z - ---- + 7 5 3 a 6 a a a a 6 6 7 7 7 3 z 21 z 2 6 4 6 7 z 10 z 5 z 7 3 7 > ---- + ----- + 5 a z - a z - ---- - ----- - ---- - 5 a z - 3 a z - 4 2 5 3 a a a a a 8 8 9 9 10 8 7 z 14 z 2 8 4 z 7 z 9 10 z > 11 z - ---- - ----- - 4 a z - ---- - ---- - 3 a z - z - --- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 2 1 5 2 8 5 8 12 + 11 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + - + 10 5 8 4 6 4 6 3 4 3 4 2 2 2 t q t q t q t q t q t q t q t 9 2 4 4 2 6 2 6 3 8 3 > ---- + 10 q t + 10 q t + 9 q t + 10 q t + 5 q t + 9 q t + 2 q t 8 4 10 4 10 5 12 5 14 6 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a40

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