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L10a121 L10a123

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

a-2z-2 + 2a-2 + a-2z2 + a-2z4 - 3z-2 - 6 - 6z2 - 3z4 - z6 + 4a2z-2 + 8a2 + 7a2z2 + 3a2z4 - 3a4z-2 - 5a4 - 3a4z2 + a6z-2 + a6

Kauffman Polynomial:

a-4z4 - a-3z3 + 4a-3z5 + a-2z-2 - 4a-2 + 6a-2z2 - 11a-2z4 + 9a-2z6 - a-1z-1 + a-1z + 4a-1z3 - 13a-1z5 + 10a-1z7 + 3z-2 - 14 + 28z2 - 30z4 + 7z6 + 5z8 - az-1 + az + 14az3 - 32az5 + 15az7 + az9 + 4a2z-2 - 21a2 + 40a2z2 - 30a2z4 - 2a2z6 + 7a2z8 - a3z-1 + a3z + 12a3z3 - 20a3z5 + 7a3z7 + a3z9 + 3a4z-2 - 14a4 + 24a4z2 - 16a4z4 + a4z6 + 2a4z8 - a5z-1 + a5z + 3a5z3 - 5a5z5 + 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                 6 1   j = 3               5 3     j = 1             9 6       j = -1           8 8         j = -3         5 6           j = -5       4 8             j = -7     2 5               j = -9   1 5                 j = -11   1                   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, 122]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 122]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 12, 17, 11], X[18, 13, 19, 14], > X[20, 18, 9, 17], X[12, 19, 13, 20], X[8, 16, 5, 15], X[14, 8, 15, 7], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 8, -7}, > {10, -2, 3, -6, 4, -8, 7, -3, 5, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-6 2 6 9 13 14 2 3 4 14 + q - -- + -- - -- + -- - -- - 11 q + 9 q - 4 q + q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 2 -16 4 4 -10 3 -6 3 2 6 8 q + --- + q + --- + --- - q + -- - q + -- + 6 q + 3 q + 3 q - 18 14 12 8 2 q q q q q 10 12 > 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 122]][a, z]
Out[8]=	2 4 6 2 2 2 4 6 3 1 4 a 3 a a 2 z -6 + -- + 8 a - 5 a + a - -- + ----- + ---- - ---- + -- - 6 z + -- + 2 2 2 2 2 2 2 2 a z a z z z z a 4 2 2 4 2 4 z 2 4 6 > 7 a z - 3 a z - 3 z + -- + 3 a z - z 2 a
In[9]:=	Kauffman[Link[10, Alternating, 122]][a, z]
Out[9]=	2 4 6 4 2 4 6 3 1 4 a 3 a a 1 a -14 - -- - 21 a - 14 a - 4 a + -- + ----- + ---- + ---- + -- - --- - - - 2 2 2 2 2 2 2 a z z a z a z z z z 3 5 2 a a z 3 5 2 6 z 2 2 4 2 > -- - -- + - + a z + a z + a z + 28 z + ---- + 40 a z + 24 a z + z z a 2 a 3 3 4 4 6 2 z 4 z 3 3 3 5 3 4 z 11 z > 6 a z - -- + ---- + 14 a z + 12 a z + 3 a z - 30 z + -- - ----- - 3 a 4 2 a a a 5 5 2 4 4 4 6 4 4 z 13 z 5 3 5 > 30 a z - 16 a z - 4 a z + ---- - ----- - 32 a z - 20 a z - 3 a a 6 7 5 5 6 9 z 2 6 4 6 6 6 10 z 7 > 5 a z + 7 z + ---- - 2 a z + a z + a z + ----- + 15 a z + 2 a a 3 7 5 7 8 2 8 4 8 9 3 9 > 7 a z + 2 a z + 5 z + 7 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 1 1 5 2 5 4 8 - + 9 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 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 > ----- + ---- + --- + 6 q t + 5 q t + 3 q t + 6 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 L10a122

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