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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 6q-15/2 + 9q-13/2 - 11q-11/2 + 11q-9/2 - 11q-7/2 + 8q-5/2 - 5q-3/2 + 2q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-30 + q-28 - q-26 + 2q-24 - q-22 - 2q-20 + q-18 - 2q-16 + 3q-14 + q-12 + 2q-10 + 3q-8 - 2q-6 + 2q-4 + q2

HOMFLY-PT Polynomial:

- 2az - az3 - 2a3z-1 - 2a3z + a3z3 + a3z5 + 4a5z-1 + 7a5z + 6a5z3 + 2a5z5 - 3a7z-1 - 6a7z - 3a7z3 + a9z-1 + a9z

Kauffman Polynomial:

- 2az + 3az3 - az5 - a2z2 + 4a2z4 - 2a2z6 - 2a3z-1 + 6a3z - 6a3z3 + 6a3z5 - 3a3z7 + 2a4 - a4z2 - 4a4z4 + 5a4z6 - 3a4z8 - 4a5z-1 + 19a5z - 31a5z3 + 21a5z5 - 6a5z7 - a5z9 + 3a6 - 6a6z2 - 4a6z4 + 10a6z6 - 6a6z8 - 3a7z-1 + 14a7z - 25a7z3 + 21a7z5 - 7a7z7 - a7z9 + 3a8 - 9a8z2 + 10a8z4 - 3a8z8 - a9z-1 + 2a9z - a9z3 + 6a9z5 - 4a9z7 + a10 - 3a10z2 + 6a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 2                     1 j = 0                   1   j = -2                 4 1   j = -4               5 2     j = -6             6 3       j = -8           5 5         j = -10         6 6           j = -12       4 6             j = -14     2 5               j = -16   1 4                 j = -18   2                   j = -20 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, 15]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 15]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 8, 17, 7], X[18, 11, 19, 12], > X[20, 13, 5, 14], X[12, 19, 13, 20], X[14, 17, 15, 18], X[8, 16, 9, 15], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -6, 5, -7, 8, -3, 7, -4, > 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 6 9 11 11 11 8 5 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 2 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 -26 2 -22 2 -18 2 3 -12 2 3 q + q - q + --- - q - --- + q - --- + --- + q + --- + -- - 24 20 16 14 10 8 q q q q q q 2 2 2 > -- + -- + q 6 4 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 15]][a, z]
Out[8]=	3 5 7 9 -2 a 4 a 3 a a 3 5 7 9 3 ----- + ---- - ---- + -- - 2 a z - 2 a z + 7 a z - 6 a z + a z - a z + z z z z 3 3 5 3 7 3 3 5 5 5 > a z + 6 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 15]][a, z]
Out[9]=	3 5 7 9 4 6 8 10 2 a 4 a 3 a a 3 5 2 a + 3 a + 3 a + a - ---- - ---- - ---- - -- - 2 a z + 6 a z + 19 a z + z z z z 7 9 11 2 2 4 2 6 2 8 2 10 2 > 14 a z + 2 a z - a z - a z - a z - 6 a z - 9 a z - 3 a z + 3 3 3 5 3 7 3 9 3 11 3 2 4 > 3 a z - 6 a z - 31 a z - 25 a z - a z + 2 a z + 4 a z - 4 4 6 4 8 4 10 4 5 3 5 5 5 > 4 a z - 4 a z + 10 a z + 6 a z - a z + 6 a z + 21 a z + 7 5 9 5 11 5 2 6 4 6 6 6 10 6 > 21 a z + 6 a z - a z - 2 a z + 5 a z + 10 a z - 3 a z - 3 7 5 7 7 7 9 7 4 8 6 8 8 8 > 3 a z - 6 a z - 7 a z - 4 a z - 3 a z - 6 a z - 3 a z - 5 9 7 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 6 6 6 5 5 6 3 5 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a15

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