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L10a153 L10a155

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-9 - 2q-8 + 7q-7 - 9q-6 + 13q-5 - 13q-4 + 13q-3 - 11q-2 + 7q-1 - 3 + q

A2 (sl(3)) Invariant:

q-30 + 3q-28 + 2q-26 + 3q-24 + 8q-22 + 3q-20 + 4q-18 + 4q-16 - 2q-14 + q-12 - 3q-10 + q-8 + q-6 - 2q-4 + 4q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + 2a2 + 2a2z2 - a2z4 - 4a4 - 5a4z2 - 3a4z4 + a6z-2 + 6a6 + 6a6z2 - 2a8z-2 - 4a8 + a10z-2

Kauffman Polynomial:

- z2 + z4 - 2az3 + 3az5 - 2a2 + 6a2z2 - 7a2z4 + 6a2z6 - 4a3z + 12a3z3 - 13a3z5 + 8a3z7 - 2a4 + 14a4z2 - 16a4z4 + a4z6 + 5a4z8 - 12a5z + 36a5z3 - 41a5z5 + 15a5z7 + a5z9 + a6z-2 - 2a6 + 6a6z2 - 5a6z4 - 11a6z6 + 8a6z8 - 2a7z-1 - 4a7z + 22a7z3 - 29a7z5 + 9a7z7 + a7z9 + 2a8z-2 - 5a8 + 5a8z2 - a8z4 - 5a8z6 + 3a8z8 - 2a9z-1 + 4a9z - 4a9z5 + 2a9z7 + a10z-2 - 4a10 + 6a10z2 - 4a10z4 + a10z6

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 = 3                     1 j = 1                   2   j = -1                 5 1   j = -3               7 3     j = -5             6 4       j = -7           7 7         j = -9         6 6           j = -11       5 9             j = -13     2 4               j = -15     5                 j = -17 1 2                   j = -19 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, 154]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 154]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 8, 15, 7], X[20, 15, 17, 16], > X[18, 11, 19, 12], X[12, 17, 13, 18], X[16, 19, 5, 20], X[8, 14, 9, 13], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {6, -5, 7, -4}, > {9, -1, 3, -8, 10, -2, 5, -6, 8, -3, 4, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-9 2 7 9 13 13 13 11 7 -3 + q - -- + -- - -- + -- - -- + -- - -- + - + q 8 7 6 5 4 3 2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 3 2 3 8 3 4 4 2 -12 3 -8 -1 + q + --- + --- + --- + --- + --- + --- + --- - --- + q - --- + q + 28 26 24 22 20 18 16 14 10 q q q q q q q q q -6 2 4 2 4 > q - -- + -- - q + q 4 2 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 154]][a, z]
Out[8]=	6 8 10 2 4 6 8 a 2 a a 2 2 2 4 2 2 a - 4 a + 6 a - 4 a + -- - ---- + --- + z + 2 a z - 5 a z + 2 2 2 z z z 6 2 2 4 4 4 > 6 a z - a z - 3 a z
In[9]:=	Kauffman[Link[10, Alternating, 154]][a, z]
Out[9]=	6 8 10 7 9 2 4 6 8 10 a 2 a a 2 a 2 a 3 -2 a - 2 a - 2 a - 5 a - 4 a + -- + ---- + --- - ---- - ---- - 4 a z - 2 2 2 z z z z z 5 7 9 2 2 2 4 2 6 2 8 2 > 12 a z - 4 a z + 4 a z - z + 6 a z + 14 a z + 6 a z + 5 a z + 10 2 3 3 3 5 3 7 3 4 2 4 > 6 a z - 2 a z + 12 a z + 36 a z + 22 a z + z - 7 a z - 4 4 6 4 8 4 10 4 5 3 5 5 5 > 16 a z - 5 a z - a z - 4 a z + 3 a z - 13 a z - 41 a z - 7 5 9 5 2 6 4 6 6 6 8 6 10 6 > 29 a z - 4 a z + 6 a z + a z - 11 a z - 5 a z + a z + 3 7 5 7 7 7 9 7 4 8 6 8 8 8 > 8 a z + 15 a z + 9 a z + 2 a z + 5 a z + 8 a z + 3 a z + 5 9 7 9 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 1 2 5 2 4 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 8 17 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 9 6 6 7 7 6 4 7 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + - + 2 q t + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 3 2 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a154

L10a153 L10a155