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DrawMorseLink

PD Presentation:

X8192 X10,3,11,4 X22,18,7,17 X16,5,17,6 X4,15,5,16 X14,22,15,21 X18,12,19,11 X20,14,21,13 X12,20,13,19 X2738 X6,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-11/2 + 2q-9/2 - 5q-7/2 + 7q-5/2 - 11q-3/2 + 12q-1/2 - 13q1/2 + 12q3/2 - 9q5/2 + 6q7/2 - 3q9/2 + q11/2

A2 (sl(3)) Invariant:

q-16 + 2q-12 + 3q-10 + 2q-8 + 5q-6 + q-2 - 3q2 + q4 - 3q6 + q8 - q12 + q14 - q16

HOMFLY-PT Polynomial:

2a-3z + 3a-3z3 + a-3z5 + 2a-1z-1 + a-1z - 4a-1z3 - 4a-1z5 - a-1z7 - 5az-1 - 11az - 10az3 - 5az5 - az7 + 3a3z-1 + 5a3z + 4a3z3 + a3z5

Kauffman Polynomial:

a-6z2 - a-6z4 + 3a-5z3 - 3a-5z5 + a-4 - 3a-4z2 + 6a-4z4 - 5a-4z6 + 2a-3z - 7a-3z3 + 9a-3z5 - 6a-3z7 - 4a-2z2 + 6a-2z6 - 5a-2z8 + 2a-1z-1 - 3a-1z - 5a-1z3 + 7a-1z5 + a-1z7 - 3a-1z9 - 5 + 19z2 - 27z4 + 21z6 - 5z8 - z10 + 5az-1 - 22az + 40az3 - 36az5 + 22az7 - 6az9 - 5a2 + 20a2z2 - 28a2z4 + 18a2z6 - 2a2z8 - a2z10 + 3a3z-1 - 15a3z + 28a3z3 - 26a3z5 + 14a3z7 - 3a3z9 + a4z2 - 8a4z4 + 8a4z6 - 2a4z8 + 2a5z - 7a5z3 + 5a5z5 - a5z7

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 12                       1 j = 10                     2   j = 8                   4 1   j = 6                 5 2     j = 4               7 4       j = 2             6 5         j = 0           7 8           j = -2         4 5             j = -4       3 7               j = -6     2 4                 j = -8     3                   j = -10 1 2                     j = -12 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, 186]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 186]]
Out[4]=	PD[X[8, 1, 9, 2], X[10, 3, 11, 4], X[22, 18, 7, 17], X[16, 5, 17, 6], > X[4, 15, 5, 16], X[14, 22, 15, 21], X[18, 12, 19, 11], X[20, 14, 21, 13], > X[12, 20, 13, 19], X[2, 7, 3, 8], X[6, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -5, 4, -11}, > {10, -1, 11, -2, 7, -9, 8, -6, 5, -4, 3, -7, 9, -8, 6, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(11/2) 2 5 7 11 12 3/2 -q + ---- - ---- + ---- - ---- + ------- - 13 Sqrt[q] + 12 q - 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 5/2 7/2 9/2 11/2 > 9 q + 6 q - 3 q + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 2 3 2 5 -2 2 4 6 8 12 14 16 q + --- + --- + -- + -- + q - 3 q + q - 3 q + q - q + q - q 12 10 8 6 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 186]][a, z]
Out[8]=	3 3 3 2 5 a 3 a 2 z z 3 3 z 4 z 3 --- - --- + ---- + --- + - - 11 a z + 5 a z + ---- - ---- - 10 a z + a z z z 3 a 3 a a a 5 5 7 3 3 z 4 z 5 3 5 z 7 > 4 a z + -- - ---- - 5 a z + a z - -- - a z 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 186]][a, z]
Out[9]=	3 -4 2 2 5 a 3 a 2 z 3 z 3 5 -5 + a - 5 a + --- + --- + ---- + --- - --- - 22 a z - 15 a z + 2 a z + a z z z 3 a a 2 2 2 3 3 3 2 z 3 z 4 z 2 2 4 2 3 z 7 z 5 z > 19 z + -- - ---- - ---- + 20 a z + a z + ---- - ---- - ---- + 6 4 2 5 3 a a a a a a 4 4 3 3 3 5 3 4 z 6 z 2 4 4 4 > 40 a z + 28 a z - 7 a z - 27 z - -- + ---- - 28 a z - 8 a z - 6 4 a a 5 5 5 6 6 3 z 9 z 7 z 5 3 5 5 5 6 5 z 6 z > ---- + ---- + ---- - 36 a z - 26 a z + 5 a z + 21 z - ---- + ---- + 5 3 a 4 2 a a a a 7 7 8 2 6 4 6 6 z z 7 3 7 5 7 8 5 z > 18 a z + 8 a z - ---- + -- + 22 a z + 14 a z - a z - 5 z - ---- - 3 a 2 a a 9 2 8 4 8 3 z 9 3 9 10 2 10 > 2 a z - 2 a z - ---- - 6 a z - 3 a z - z - a z a
In[10]:=	Kh[L][q, t]
Out[10]=	2 1 1 2 3 2 4 3 7 8 + 6 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + 12 6 10 6 10 5 8 4 6 4 6 3 4 3 4 2 q t q t q t q t q t q t q t q t 4 7 5 2 4 4 2 6 2 6 3 > ----- + - + ---- + 5 q t + 7 q t + 4 q t + 5 q t + 2 q t + 2 2 t 2 q t q t 8 3 8 4 10 4 12 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a186

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