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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,11,19,12 X20,17,5,18 X12,19,13,20 X16,10,17,9 X2,14,3,13

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 5q-11/2 - 9q-9/2 + 13q-7/2 - 17q-5/2 + 17q-3/2 - 17q-1/2 + 12q1/2 - 8q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

q-20 - 2q-18 - 2q-16 + q-14 - 4q-12 + 3q-10 + 2q-8 + 2q-6 + 6q-4 - q-2 + 5 - 2q2 - q4 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 - 2az-1 + az + 6az3 + 4az5 + az7 + 3a3z-1 - 4a3z3 - 2a3z5 - a5z-1 + a5z3

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 6a-2z4 - 4a-2z6 + 2a-1z - 8a-1z3 + 12a-1z5 - 7a-1z7 - 5z2 + 7z4 + 3z6 - 6z8 + 2az-1 + 3az - 26az3 + 37az5 - 15az7 - 2az9 - 3a2 - 5a2z2 + 5a2z4 + 15a2z6 - 13a2z8 + 3a3z-1 + a3z - 23a3z3 + 40a3z5 - 17a3z7 - 2a3z9 - 3a4 - 2a4z2 + 10a4z4 + 3a4z6 - 7a4z8 + a5z-1 - 6a5z3 + 15a5z5 - 9a5z7 - a6 + 6a6z4 - 5a6z6 - 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                   3   j = 4                 5 1   j = 2               7 3     j = 0             10 5       j = -2           9 9         j = -4         8 8           j = -6       5 9             j = -8     4 8               j = -10   1 5                 j = -12   4                   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, 1]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 1]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 6, 11, 5], > X[8, 4, 9, 3], X[18, 11, 19, 12], X[20, 17, 5, 18], X[12, 19, 13, 20], > X[16, 10, 17, 9], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 5, -3}, {4, -1, 2, -5, 9, -4, 6, -8, 10, -2, 3, -9, 7, -6, > 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 5 9 13 17 17 17 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 12 Sqrt[q] - 8 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 2 -14 4 3 2 2 6 -2 2 4 5 + q - --- - --- + q - --- + --- + -- + -- + -- - q - 2 q - q + 18 16 12 10 8 6 4 q q q q q q q 6 8 10 > 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 1]][a, z]
Out[8]=	3 5 3 5 -2 a 3 a a z 2 z 3 3 3 5 3 z 5 ---- + ---- - -- - - + a z - ---- + 6 a z - 4 a z + a z - -- + 4 a z - z z z a a a 3 5 7 > 2 a z + a z
In[9]:=	Kauffman[Link[10, Alternating, 1]][a, z]
Out[9]=	3 5 2 2 4 6 2 a 3 a a 2 z 3 2 2 z -3 a - 3 a - a + --- + ---- + -- + --- + 3 a z + a z - 5 z - ---- - z z z a 2 a 3 3 2 2 4 2 z 8 z 3 3 3 5 3 4 > 5 a z - 2 a z + -- - ---- - 26 a z - 23 a z - 6 a z + 7 z + 3 a a 4 5 5 6 z 2 4 4 4 6 4 z 12 z 5 3 5 > ---- + 5 a z + 10 a z + 6 a z - -- + ----- + 37 a z + 40 a z + 2 3 a a a 6 7 5 5 7 5 6 4 z 2 6 4 6 6 6 7 z > 15 a z - a z + 3 z - ---- + 15 a z + 3 a z - 5 a z - ---- - 2 a a 7 3 7 5 7 8 2 8 4 8 9 3 9 > 15 a z - 17 a z - 9 a z - 6 z - 13 a z - 7 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 4 1 5 4 8 5 9 10 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 8 8 9 2 2 2 4 2 4 3 6 3 > ----- + ---- + ---- + 5 t + 7 q t + 3 q t + 5 q t + q t + 3 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 L10a1

L9n28 L10a2