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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X18,8,19,7 X20,15,5,16 X16,19,17,20 X14,10,15,9 X10,14,11,13 X8,18,9,17 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 5q-9/2 + 8q-7/2 - 11q-5/2 + 11q-3/2 - 11q-1/2 + 9q1/2 - 6q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + q-16 + 2q-14 - 3q-12 + q-10 + q-8 + 3q-4 - q-2 + 3 - q2 - q4 + 2q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z + a-1z + 2a-1z3 - az-1 - 2az - az3 - az5 + 2a3z-1 + 3a3z + 3a3z3 - 2a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - 3a-2z2 + 6a-2z4 - 3a-2z6 - a-1z3 + 6a-1z5 - 4a-1z7 - z2 + 4z4 + z6 - 3z8 + az-1 - 2az - az3 + 8az5 - 5az7 - az9 - a2z4 + 5a2z6 - 5a2z8 + 2a3z-1 - 8a3z + 8a3z3 + a3z5 - 3a3z7 - a3z9 + a4 - 4a4z2 + 5a4z4 - a4z6 - 2a4z8 + 2a5z-1 - 8a5z + 9a5z3 - a5z5 - 2a5z7 - 2a6z2 + 4a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

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 j = 8                     1 j = 6                   2   j = 4                 4 1   j = 2               5 2     j = 0             6 4       j = -2           6 6         j = -4         5 5           j = -6       3 6             j = -8     2 5               j = -10   1 4                 j = -12   1                   j = -14 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[10, Alternating, 37]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 37]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[18, 8, 19, 7], X[20, 15, 5, 16], > X[16, 19, 17, 20], X[14, 10, 15, 9], X[10, 14, 11, 13], X[8, 18, 9, 17], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 10, -2, 7, -6, 4, -5, 8, -3, > 5, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 5 8 11 11 11 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 9 Sqrt[q] - 6 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 -16 2 3 -10 -8 3 -2 2 4 6 3 + q + --- + q + --- - --- + q + q + -- - q - q - q + 2 q - 20 14 12 4 q q q q 8 12 > 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 37]][a, z]
Out[8]=	3 5 7 3 a 2 a 2 a a z z 3 5 2 z 3 -(-) + ---- - ---- + -- - -- + - - 2 a z + 3 a z - 3 a z + ---- - a z + z z z z 3 a a a 3 3 5 > 3 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 37]][a, z]
Out[9]=	3 5 7 2 4 a 2 a 2 a a z 3 5 7 2 3 z a + - + ---- + ---- + -- - -- - 2 a z - 8 a z - 8 a z - 3 a z - z - ---- - z z z z 3 2 a a 3 3 4 2 6 2 2 z z 3 3 3 5 3 7 3 4 > 4 a z - 2 a z + ---- - -- - a z + 8 a z + 9 a z + 3 a z + 4 z + 3 a a 4 5 5 6 z 2 4 4 4 6 4 z 6 z 5 3 5 5 5 > ---- - a z + 5 a z + 4 a z - -- + ---- + 8 a z + a z - a z - 2 3 a a a 6 7 7 5 6 3 z 2 6 4 6 6 6 4 z 7 3 7 > a z + z - ---- + 5 a z - a z - 2 a z - ---- - 5 a z - 3 a z - 2 a a 5 7 8 2 8 4 8 9 3 9 > 2 a z - 3 z - 5 a z - 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 1 4 2 5 3 6 6 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 5 5 6 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 4 t + 5 q t + 2 q t + 4 q t + q t + 2 q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a37

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