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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-17/2 - 2q-15/2 + 5q-13/2 - 8q-11/2 + 10q-9/2 - 11q-7/2 + 9q-5/2 - 9q-3/2 + 5q-1/2 - 3q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-26 - q-24 - q-22 - 3q-20 + q-18 - q-16 + q-14 + 3q-12 + q-10 + 6q-8 + q-6 + 3q-4 + q-2 - 1 + q2 - q4

HOMFLY-PT Polynomial:

2az + 3az3 + az5 - 3a3z-1 - 10a3z - 10a3z3 - 5a3z5 - a3z7 + 5a5z-1 + 11a5z + 8a5z3 + 2a5z5 - 2a7z-1 - 3a7z - a7z3

Kauffman Polynomial:

- 2z2 + 3z4 - z6 + 4az - 10az3 + 10az5 - 3az7 - 4a2z2 + 2a2z4 + 6a2z6 - 3a2z8 - 3a3z-1 + 16a3z - 29a3z3 + 26a3z5 - 6a3z7 - a3z9 + 5a4 - 11a4z2 + 2a4z4 + 11a4z6 - 6a4z8 - 5a5z-1 + 17a5z - 29a5z3 + 24a5z5 - 7a5z7 - a5z9 + 5a6 - 11a6z2 + 6a6z4 + a6z6 - 3a6z8 - 2a7z-1 + 5a7z - 8a7z3 + 6a7z5 - 4a7z7 + 2a8z4 - 3a8z6 + 2a9z3 - 2a9z5 - a10 + 2a10z2 - a10z4

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 4                     1 j = 2                   2   j = 0                 3 1   j = -2               6 2     j = -4             5 5       j = -6           6 4         j = -8         4 5           j = -10       4 6             j = -12     1 4               j = -14   1 4                 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[10, Alternating, 18]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 18]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 4, 11, 3], X[12, 10, 13, 9], X[18, 13, 19, 14], > X[16, 7, 17, 8], X[8, 17, 9, 18], X[20, 15, 5, 16], X[14, 19, 15, 20], > X[2, 5, 3, 6], X[4, 12, 1, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 5, -6, 3, -2, 10, -3, 4, -8, 7, -5, 6, -4, > 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 2 5 8 10 11 9 9 5 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 > 3 Sqrt[q] + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 -24 -22 3 -18 -16 -14 3 -10 6 -6 -1 - q - q - q - --- + q - q + q + --- + q + -- + q + 20 12 8 q q q 3 -2 2 4 > -- + q + q - q 4 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 18]][a, z]
Out[8]=	3 5 7 -3 a 5 a 2 a 3 5 7 3 3 3 ----- + ---- - ---- + 2 a z - 10 a z + 11 a z - 3 a z + 3 a z - 10 a z + z z z 5 3 7 3 5 3 5 5 5 3 7 > 8 a z - a z + a z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 18]][a, z]
Out[9]=	3 5 7 4 6 10 3 a 5 a 2 a 3 5 7 5 a + 5 a - a - ---- - ---- - ---- + 4 a z + 16 a z + 17 a z + 5 a z - z z z 2 2 2 4 2 6 2 10 2 3 3 3 > 2 z - 4 a z - 11 a z - 11 a z + 2 a z - 10 a z - 29 a z - 5 3 7 3 9 3 4 2 4 4 4 6 4 > 29 a z - 8 a z + 2 a z + 3 z + 2 a z + 2 a z + 6 a z + 8 4 10 4 5 3 5 5 5 7 5 9 5 6 > 2 a z - a z + 10 a z + 26 a z + 24 a z + 6 a z - 2 a z - z + 2 6 4 6 6 6 8 6 7 3 7 5 7 > 6 a z + 11 a z + a z - 3 a z - 3 a z - 6 a z - 7 a z - 7 7 2 8 4 8 6 8 3 9 5 9 > 4 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 1 1 1 4 1 4 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 4 5 6 4 5 2 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + t + 2 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 4 3 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a18

L10a17 L10a19