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DrawMorseLink

PD Presentation:

X6172 X14,6,15,5 X10,4,11,3 X2,16,3,15 X18,7,19,8 X16,9,17,10 X8,17,9,18 X20,12,13,11 X12,14,5,13 X4,19,1,20

Gauss Code:

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

Jones Polynomial:

- q-4 + 3q-3 - 4q-2 + 8q-1 - 9 + 11q - 9q2 + 9q3 - 6q4 + 3q5 - q6

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 + 2q-6 + 5q-4 + 2q-2 + 5 + 4q2 + 3q4 + 5q6 + 2q10 - q12 - q14 + q16 - q18

HOMFLY-PT Polynomial:

- a-4 - 2a-4z2 - a-4z4 + a-2z-2 + 3a-2 + 3a-2z2 + 3a-2z4 + a-2z6 - 2z-2 - 3 + z2 + 3z4 + z6 + a2z-2 + a2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-7z3 + 3a-6z4 + a-5z - 4a-5z3 + 6a-5z5 - 2a-4 + 9a-4z2 - 15a-4z4 + 9a-4z6 + 3a-3z + a-3z3 - 13a-3z5 + 8a-3z7 + a-2z-2 - 6a-2 + 13a-2z2 - 12a-2z4 - 5a-2z6 + 5a-2z8 - 2a-1z-1 + 3a-1z + 7a-1z3 - 16a-1z5 + 2a-1z7 + 2a-1z9 + 2z-2 - 5 - 3z2 + 26z4 - 28z6 + 8z8 - 2az-1 + az + 5az3 - az5 - 5az7 + 2az9 + a2z-2 - 2a2 - 7a2z2 + 20a2z4 - 14a2z6 + 3a2z8 + 4a3z3 - 4a3z5 + a3z7

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 = 13                     1 j = 11                   2   j = 9                 4 1   j = 7               5 2     j = 5             4 4       j = 3           7 5         j = 1         6 8           j = -1       2 3             j = -3     2 6               j = -5   1 2                 j = -7   2                   j = -9 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, 141]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 141]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 6, 15, 5], X[10, 4, 11, 3], X[2, 16, 3, 15], > X[18, 7, 19, 8], X[16, 9, 17, 10], X[8, 17, 9, 18], X[20, 12, 13, 11], > X[12, 14, 5, 13], X[4, 19, 1, 20]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {2, -1, 5, -7, 6, -3, 8, -9}, > {9, -2, 4, -6, 7, -5, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 4 8 2 3 4 5 6 -9 - q + -- - -- + - + 11 q - 9 q + 9 q - 6 q + 3 q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 -8 2 5 2 2 4 6 10 12 14 5 - q + q + q + -- + -- + -- + 4 q + 3 q + 5 q + 2 q - q - q + 6 4 2 q q q 16 18 > q - q
In[8]:=	HOMFLYPT[Link[10, Alternating, 141]][a, z]
Out[8]=	2 2 2 4 -4 3 2 2 1 a 2 2 z 3 z 2 2 4 z -3 - a + -- + a - -- + ----- + -- + z - ---- + ---- - 2 a z + 3 z - -- + 2 2 2 2 2 4 2 4 a z a z z a a a 4 6 3 z 2 4 6 z > ---- - a z + z + -- 2 2 a a
In[9]:=	Kauffman[Link[10, Alternating, 141]][a, z]
Out[9]=	2 2 6 2 2 1 a 2 2 a z 3 z 3 z -5 - -- - -- - 2 a + -- + ----- + -- - --- - --- + -- + --- + --- + a z - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 3 3 3 3 2 9 z 13 z 2 2 z 4 z z 7 z 3 3 3 > 3 z + ---- + ----- - 7 a z + -- - ---- + -- + ---- + 5 a z + 4 a z + 4 2 7 5 3 a a a a a a 4 4 4 5 5 5 4 3 z 15 z 12 z 2 4 6 z 13 z 16 z 5 > 26 z + ---- - ----- - ----- + 20 a z + ---- - ----- - ----- - a z - 6 4 2 5 3 a a a a a a 6 6 7 7 3 5 6 9 z 5 z 2 6 8 z 2 z 7 3 7 > 4 a z - 28 z + ---- - ---- - 14 a z + ---- + ---- - 5 a z + a z + 4 2 3 a a a a 8 9 8 5 z 2 8 2 z 9 > 8 z + ---- + 3 a z + ---- + 2 a z 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 2 2 6 2 3 6 q 8 q + 7 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 5 q t + 4 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 11 4 13 5 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a141

L10a140 L10a142