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L10a133 L10a135

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-6 - 2q-5 + 7q-4 - 9q-3 + 13q-2 - 14q-1 + 14 - 11q + 8q2 - 4q3 + q4

A2 (sl(3)) Invariant:

q-20 + 2q-18 + q-16 + 5q-14 + 6q-12 + 2q-10 + 6q-8 + q-6 + q-4 + 2q-2 - 2 + 3q2 - 3q4 + q6 + 2q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2 + a-2z2 + a-2z4 - 2 - 5z2 - 3z4 - z6 + a2z-2 + 4a2 + 6a2z2 + 3a2z4 - 2a4z-2 - 4a4 - 3a4z2 + a6z-2 + a6

Kauffman Polynomial:

a-4z4 - 2a-3z3 + 4a-3z5 - a-2 + 4a-2z2 - 9a-2z4 + 8a-2z6 - 2a-1z + 5a-1z3 - 11a-1z5 + 9a-1z7 - 1 + 13z2 - 22z4 + 5z6 + 5z8 - 6az + 27az3 - 39az5 + 16az7 + az9 + a2z-2 - 3a2 + 9a2z2 - 9a2z4 - 9a2z6 + 8a2z8 - 2a3z-1 + 20a3z3 - 28a3z5 + 9a3z7 + a3z9 + 2a4z-2 - 6a4 + 6a4z2 - a4z4 - 5a4z6 + 3a4z8 - 2a5z-1 + 4a5z - 4a5z5 + 2a5z7 + a6z-2 - 4a6 + 6a6z2 - 4a6z4 + 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                 5 1   j = 3               6 3     j = 1             8 5       j = -1           8 8         j = -3         5 6           j = -5       5 9             j = -7     2 4               j = -9     5                 j = -11 1 2                   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, 134]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 134]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[16, 7, 17, 8], X[20, 9, 11, 10], > X[18, 12, 19, 11], X[8, 15, 9, 16], X[10, 19, 5, 20], X[14, 18, 15, 17], > X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -6, 4, -7}, > {5, -2, 10, -8, 6, -3, 8, -5, 7, -4}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 7 9 13 14 2 3 4 14 + q - -- + -- - -- + -- - -- - 11 q + 8 q - 4 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 5 6 2 6 -6 -4 2 2 4 -2 + q + --- + q + --- + --- + --- + -- + q + q + -- + 3 q - 3 q + 18 14 12 10 8 2 q q q q q q 6 8 10 12 > q + 2 q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 134]][a, z]
Out[8]=	2 4 6 2 -2 2 4 6 a 2 a a 2 z 2 2 4 2 -2 + a + 4 a - 4 a + a + -- - ---- + -- - 5 z + -- + 6 a z - 3 a z - 2 2 2 2 z z z a 4 4 z 2 4 6 > 3 z + -- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 134]][a, z]
Out[9]=	2 4 6 3 5 -2 2 4 6 a 2 a a 2 a 2 a 2 z -1 - a - 3 a - 6 a - 4 a + -- + ---- + -- - ---- - ---- - --- - 6 a z + 2 2 2 z z a z z z 2 3 3 5 2 4 z 2 2 4 2 6 2 2 z 5 z > 4 a z + 13 z + ---- + 9 a z + 6 a z + 6 a z - ---- + ---- + 2 3 a a a 4 4 5 3 3 3 4 z 9 z 2 4 4 4 6 4 4 z > 27 a z + 20 a z - 22 z + -- - ---- - 9 a z - a z - 4 a z + ---- - 4 2 3 a a a 5 6 11 z 5 3 5 5 5 6 8 z 2 6 4 6 > ----- - 39 a z - 28 a z - 4 a z + 5 z + ---- - 9 a z - 5 a z + a 2 a 7 6 6 9 z 7 3 7 5 7 8 2 8 4 8 > a z + ---- + 16 a z + 9 a z + 2 a z + 5 z + 8 a z + 3 a z + a 9 3 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 1 2 5 2 4 5 9 - + 8 q + ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + q 13 6 11 6 11 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 5 6 8 3 3 2 5 2 5 3 7 3 > ----- + ---- + --- + 5 q t + 6 q t + 3 q t + 5 q t + q t + 3 q t + 3 2 3 q t q t q t 9 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a134

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