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Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - q-5 + 5q-4 - 6q-3 + 11q-2 - 11q-1 + 11 - 10q + 7q2 - 4q3 + q4

A2 (sl(3)) Invariant:

q-20 + 3q-18 + 4q-16 + 7q-14 + 8q-12 + 4q-10 + 6q-8 + q-6 - q-4 - 4 + q2 - 3q4 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 - z-2 - 3 - 4z2 - 3z4 - z6 + 4a2z-2 + 10a2 + 9a2z2 + 3a2z4 - 5a4z-2 - 8a4 - 3a4z2 + 2a6z-2 + a6

Kauffman Polynomial:

a-4z4 - 3a-3z3 + 4a-3z5 + a-2z2 - 8a-2z4 + 7a-2z6 - a-1z-1 + 3a-1z - 8a-1z5 + 7a-1z7 + z-2 - 2 + z2 - 3z4 - z6 + 4z8 - 5az-1 + 13az - 8az3 - 8az5 + 6az7 + az9 + 4a2z-2 - 10a2 + 11a2z2 - 3a2z4 - 7a2z6 + 5a2z8 - 9a3z-1 + 21a3z - 17a3z3 + 3a3z5 + a3z9 + 5a4z-2 - 14a4 + 20a4z2 - 14a4z4 + 2a4z6 + a4z8 - 5a5z-1 + 11a5z - 6a5z3 - a5z5 + a5z7 + 2a6z-2 - 7a6 + 9a6z2 - 5a6z4 + a6z6

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 = 9                     1 j = 7                   3   j = 5                 4 1   j = 3               6 3     j = 1             5 4       j = -1           7 7         j = -3         4 4           j = -5       2 7             j = -7     3 4               j = -9   1 5                 j = -11                       j = -13 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]=	3
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 125]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 125]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 7, 17, 8], X[8, 15, 5, 16], > X[20, 14, 9, 13], X[18, 12, 19, 11], X[12, 20, 13, 19], X[14, 18, 15, 17], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -4}, > {10, -2, 6, -7, 5, -8, 4, -3, 8, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 -5 5 6 11 11 2 3 4 11 + q - q + -- - -- + -- - -- - 10 q + 7 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 3 4 7 8 4 6 -6 -4 2 4 8 -4 + q + --- + --- + --- + --- + --- + -- + q - q + q - 3 q + q - 18 16 14 12 10 8 q q q q q q 10 12 > 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 125]][a, z]
Out[8]=	2 4 6 2 2 4 6 -2 4 a 5 a 2 a 2 z 2 2 -3 + 10 a - 8 a + a - z + ---- - ---- + ---- - 4 z + -- + 9 a z - 2 2 2 2 z z z a 4 4 2 4 z 2 4 6 > 3 a z - 3 z + -- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 125]][a, z]
Out[9]=	2 4 6 3 2 4 6 -2 4 a 5 a 2 a 1 5 a 9 a -2 - 10 a - 14 a - 7 a + z + ---- + ---- + ---- - --- - --- - ---- - 2 2 2 a z z z z z z 5 2 5 a 3 z 3 5 2 z 2 2 4 2 > ---- + --- + 13 a z + 21 a z + 11 a z + z + -- + 11 a z + 20 a z + z a 2 a 3 4 4 6 2 3 z 3 3 3 5 3 4 z 8 z 2 4 > 9 a z - ---- - 8 a z - 17 a z - 6 a z - 3 z + -- - ---- - 3 a z - 3 4 2 a a a 5 5 6 4 4 6 4 4 z 8 z 5 3 5 5 5 6 7 z > 14 a z - 5 a z + ---- - ---- - 8 a z + 3 a z - a z - z + ---- - 3 a 2 a a 7 2 6 4 6 6 6 7 z 7 5 7 8 2 8 > 7 a z + 2 a z + a z + ---- + 6 a z + a z + 4 z + 5 a z + a 4 8 9 3 9 > a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 1 1 5 3 4 2 7 4 - + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t q t 4 7 3 3 2 5 2 5 3 7 3 9 4 > ---- + --- + 4 q t + 6 q t + 3 q t + 4 q t + q t + 3 q t + q t 3 q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a125

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