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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 4q-4 - 8q-3 + 13q-2 - 15q-1 + 18 - 15q + 13q2 - 8q3 + 4q4 - q5

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 2q-10 + 3q-8 + 2q-6 + q-4 + 8q-2 + 1 + 8q2 + q4 + 2q6 + 3q8 - 2q10 + 2q12 - q14

HOMFLY-PT Polynomial:

a-2z-2 + a-2 - 2a-2z2 - 3a-2z4 - a-2z6 - 2z-2 - 2 + 3z2 + 8z4 + 5z6 + z8 + a2z-2 + a2 - 2a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 + 4a-3z3 - 11a-3z5 + 7a-3z7 + a-2z-2 - 2a-2 + a-2z2 + 2a-2z4 - 9a-2z6 + 7a-2z8 - 2a-1z-1 + 2a-1z + 9a-1z3 - 19a-1z5 + 7a-1z7 + 3a-1z9 + 2z-2 - 3 - 2z2 + 16z4 - 26z6 + 14z8 - 2az-1 + 2az + 9az3 - 19az5 + 7az7 + 3az9 + a2z-2 - 2a2 + a2z2 + 2a2z4 - 9a2z6 + 7a2z8 + 4a3z3 - 11a3z5 + 7a3z7 + 2a4z2 - 6a4z4 + 4a4z6 - a5z3 + a5z5

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 j = 11                     1 j = 9                   3   j = 7                 5 1   j = 5               8 3     j = 3             8 6       j = 1           10 7         j = -1         7 10           j = -3       6 8             j = -5     3 8               j = -7   1 5                 j = -9   3                   j = -11 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 156]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 156]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 10, 19, 9], X[6, 18, 1, 17], X[16, 7, 17, 8], > X[10, 4, 11, 3], X[14, 6, 15, 5], X[4, 14, 5, 13], X[20, 11, 13, 12], > X[12, 15, 7, 16], X[2, 19, 3, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -7, 6, -3}, {4, -1, 2, -5, 8, -9}, > {7, -6, 9, -4, 3, -2, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-5 4 8 13 15 2 3 4 5 18 - q + -- - -- + -- - -- - 15 q + 13 q - 8 q + 4 q - q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 3 2 -4 8 2 4 6 8 10 1 - q + --- - --- + -- + -- + q + -- + 8 q + q + 2 q + 3 q - 2 q + 12 10 8 6 2 q q q q q 12 14 > 2 q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 156]][a, z]
Out[8]=	2 2 4 -2 2 2 1 a 2 2 z 2 2 4 3 z -2 + a + a - -- + ----- + -- + 3 z - ---- - 2 a z + 8 z - ---- - 2 2 2 2 2 2 z a z z a a 6 2 4 6 z 2 6 8 > 3 a z + 5 z - -- - a z + z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 156]][a, z]
Out[9]=	2 2 2 2 2 2 1 a 2 2 a 2 z 2 2 z z -3 - -- - 2 a + -- + ----- + -- - --- - --- + --- + 2 a z - 2 z + ---- + -- + 2 2 2 2 2 a z z a 4 2 a z a z z a a 3 3 3 2 2 4 2 z 4 z 9 z 3 3 3 5 3 4 > a z + 2 a z - -- + ---- + ---- + 9 a z + 4 a z - a z + 16 z - 5 3 a a a 4 4 5 5 5 6 z 2 z 2 4 4 4 z 11 z 19 z 5 3 5 > ---- + ---- + 2 a z - 6 a z + -- - ----- - ----- - 19 a z - 11 a z + 4 2 5 3 a a a a a 6 6 7 7 5 5 6 4 z 9 z 2 6 4 6 7 z 7 z 7 > a z - 26 z + ---- - ---- - 9 a z + 4 a z + ---- + ---- + 7 a z + 4 2 3 a a a a 8 9 3 7 8 7 z 2 8 3 z 9 > 7 a z + 14 z + ---- + 7 a z + ---- + 3 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 3 1 5 3 8 6 8 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 7 3 3 2 5 2 5 3 7 3 7 4 > --- + 7 q t + 8 q t + 6 q t + 8 q t + 3 q t + 5 q t + q t + q t 9 4 11 5 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a156

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