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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X14,11,5,12 X12,7,13,8 X8,13,9,14 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 3q-13/2 - 4q-11/2 + 3q-9/2 - 4q-7/2 + 2q-5/2 - q-3/2

A2 (sl(3)) Invariant:

- q-28 - 2q-26 + q-20 + 3q-18 + 2q-16 + 3q-14 + q-12 + q-10 + q-8 - q-6 + q-4

HOMFLY-PT Polynomial:

- a3z - a3z3 - 2a5z-1 - 4a5z - 2a5z3 + 3a7z-1 + 3a7z - a9z-1

Kauffman Polynomial:

a3z - a3z3 + a4z2 - 2a4z4 + 2a5z-1 - 5a5z + 5a5z3 - 3a5z5 - 3a6 + 6a6z2 - 3a6z4 - a6z6 + 3a7z-1 - 8a7z + 10a7z3 - 5a7z5 - 3a8 + 7a8z2 - 2a8z4 - a8z6 + a9z-1 - 2a9z + 4a9z3 - 2a9z5 - a10 + 2a10z2 - a10z4

Khovanov Homology:

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

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L7a2

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