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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X14,12,15,11 X10,4,11,3 X12,5,13,6 X20,13,5,14 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 7q-13/2 - 10q-11/2 + 12q-9/2 - 14q-7/2 + 12q-5/2 - 10q-3/2 + 6q-1/2 - 4q1/2 + q3/2

A2 (sl(3)) Invariant:

- q-26 - 3q-20 + q-18 - 2q-16 + q-14 + 3q-12 + 5q-8 - q-6 + 3q-4 + 2q-2 + 2q2 - q4

HOMFLY-PT Polynomial:

- az-1 - az + 2az3 + az5 - 2a3z - 5a3z3 - 4a3z5 - a3z7 + 2a5z-1 + 5a5z + 6a5z3 + 2a5z5 - a7z-1 - 2a7z - a7z3

Kauffman Polynomial:

2z4 - z6 + az-1 - az - 8az3 + 12az5 - 4az7 - a2 + 2a2z2 - 10a2z4 + 14a2z6 - 5a2z8 + 4a3z - 18a3z3 + 22a3z5 - 4a3z7 - 2a3z9 + 3a4 - 3a4z2 - 12a4z4 + 23a4z6 - 10a4z8 - 2a5z-1 + 9a5z - 20a5z3 + 23a5z5 - 7a5z7 - 2a5z9 + 5a6 - 12a6z2 + 9a6z4 + 2a6z6 - 5a6z8 - a7z-1 + 4a7z - 8a7z3 + 10a7z5 - 7a7z7 + 2a8 - 6a8z2 + 8a8z4 - 6a8z6 + 2a9z3 - 3a9z5 + a10z2 - 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                   3   j = 0                 3 1   j = -2               7 3     j = -4             7 5       j = -6           7 5         j = -8         5 7           j = -10       5 7             j = -12     2 5               j = -14   1 5                 j = -16   2                   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, 23]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 23]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[18, 9, 19, 10], X[8, 17, 9, 18], > X[4, 19, 1, 20], X[14, 12, 15, 11], X[10, 4, 11, 3], X[12, 5, 13, 6], > X[20, 13, 5, 14], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 7, -5}, {8, -1, 2, -4, 3, -7, 6, -8, 9, -6, 10, -2, 4, -3, > 5, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 7 10 12 14 12 10 6 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 > 4 Sqrt[q] + q
In[7]:=	A2Invariant[L][q]
Out[7]=	-26 3 -18 2 -14 3 5 -6 3 2 2 4 -q - --- + q - --- + q + --- + -- - q + -- + -- + 2 q - q 20 16 12 8 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 23]][a, z]
Out[8]=	5 7 a 2 a a 3 5 7 3 3 3 -(-) + ---- - -- - a z - 2 a z + 5 a z - 2 a z + 2 a z - 5 a z + z z z 5 3 7 3 5 3 5 5 5 3 7 > 6 a z - a z + a z - 4 a z + 2 a z - a z
In[9]:=	Kauffman[Link[10, Alternating, 23]][a, z]
Out[9]=	5 7 2 4 6 8 a 2 a a 3 5 7 -a + 3 a + 5 a + 2 a + - - ---- - -- - a z + 4 a z + 9 a z + 4 a z + z z z 2 2 4 2 6 2 8 2 10 2 3 3 3 > 2 a z - 3 a z - 12 a z - 6 a z + a z - 8 a z - 18 a z - 5 3 7 3 9 3 4 2 4 4 4 6 4 > 20 a z - 8 a z + 2 a z + 2 z - 10 a z - 12 a z + 9 a z + 8 4 10 4 5 3 5 5 5 7 5 9 5 > 8 a z - a z + 12 a z + 22 a z + 23 a z + 10 a z - 3 a z - 6 2 6 4 6 6 6 8 6 7 3 7 5 7 > z + 14 a z + 23 a z + 2 a z - 6 a z - 4 a z - 4 a z - 7 a z - 7 7 2 8 4 8 6 8 3 9 5 9 > 7 a z - 5 a z - 10 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 2 1 5 2 5 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 5 7 7 5 7 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + t + 3 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 L10a23

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