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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-4 - 2q-3 + 5q-2 - 6q-1 + 8 - 7q + 7q2 - 4q3 + 3q4 - q5

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + 2q-8 + 2q-6 + q-4 + 5q-2 + 3 + 5q2 + 3q4 + 2q6 + 3q8 - q10 + q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z-2 + 2a-2 + a-2z2 + a-2z4 - 2z-2 - 3 - z2 + z4 + a2z-2 - 2a2z2 + a4

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 5a-4z2 - 8a-4z4 + 3a-4z6 + 2a-3z3 - 6a-3z5 + 3a-3z7 + a-2z-2 - 5a-2 + 11a-2z2 - 13a-2z4 + 4a-2z6 + a-2z8 - 2a-1z-1 + 6a-1z - a-1z3 - 7a-1z5 + 5a-1z7 + 2z-2 - 8 + 12z2 - 10z4 + 4z6 + z8 - 2az-1 + 6az - 7az3 + 2az5 + 2az7 + a2z-2 - 3a2 + 4a2z2 - 4a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

Khovanov Homology:

trqj 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                 2   j = 7               2 1   j = 5             5 2     j = 3           4 4       j = 1         4 3         j = -1       3 5           j = -3     2 3             j = -5     3               j = -7 1 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[9, Alternating, 49]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 49]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[16, 10, 17, 9], X[14, 8, 15, 7], > X[18, 14, 11, 13], X[10, 16, 5, 15], X[8, 18, 9, 17], X[2, 5, 3, 6], > X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -8, 2, -9}, {8, -1, 4, -7, 3, -6}, {9, -2, 5, -4, 6, -3, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-4 2 5 6 2 3 4 5 8 + q - -- + -- - - - 7 q + 7 q - 4 q + 3 q - q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 -12 -10 2 2 -4 5 2 4 6 8 10 3 + q + q - q + -- + -- + q + -- + 5 q + 3 q + 2 q + 3 q - q + 8 6 2 q q q 12 14 16 > q + q - q
In[8]:=	HOMFLYPT[Link[9, Alternating, 49]][a, z]
Out[8]=	2 2 2 4 2 4 2 1 a 2 z z 2 2 4 z -3 + -- + a - -- + ----- + -- - z - -- + -- - 2 a z + z + -- 2 2 2 2 2 4 2 2 a z a z z a a a
In[9]:=	Kauffman[Link[9, Alternating, 49]][a, z]
Out[9]=	2 5 2 4 2 1 a 2 2 a 6 z 2 -8 - -- - 3 a + a + -- + ----- + -- - --- - --- + --- + 6 a z + 12 z + 2 2 2 2 2 a z z a a z a z z 2 2 3 3 3 5 z 11 z 2 2 4 2 2 z 2 z z 3 3 3 > ---- + ----- + 4 a z - 2 a z - ---- + ---- - -- - 7 a z - 2 a z - 4 2 5 3 a a a a a 4 4 5 5 5 4 8 z 13 z 2 4 4 4 z 6 z 7 z 5 > 10 z - ---- - ----- - 4 a z + a z + -- - ---- - ---- + 2 a z + 4 2 5 3 a a a a a 6 6 7 7 8 3 5 6 3 z 4 z 2 6 3 z 5 z 7 8 z > 2 a z + 4 z + ---- + ---- + 3 a z + ---- + ---- + 2 a z + z + -- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	5 1 1 2 3 2 3 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 3 q t + 4 q t + q 9 4 7 4 7 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 4 q t + 5 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L9a49

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