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L10a152 L10a154

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-13 - 2q-12 + 5q-11 - 7q-10 + 9q-9 - 9q-8 + 9q-7 - 6q-6 + 5q-5 - 2q-4 + q-3

A2 (sl(3)) Invariant:

q-40 + 2q-38 + q-36 + 3q-34 + q-32 + 2q-30 + 3q-28 + q-26 + 5q-24 + q-22 + 3q-20 + 2q-18 + 2q-14 - q-12 + q-10

HOMFLY-PT Polynomial:

a6 + 4a6z2 + 4a6z4 + a6z6 + a8z-2 + 6a8 + 12a8z2 + 9a8z4 + 2a8z6 - 2a10z-2 - 10a10 - 11a10z2 - 3a10z4 + a12z-2 + 3a12 + a12z2

Kauffman Polynomial:

- a6 + 4a6z2 - 4a6z4 + a6z6 + 3a7z3 - 6a7z5 + 2a7z7 - a8z-2 + 7a8 - 14a8z2 + 15a8z4 - 11a8z6 + 3a8z8 + 2a9z-1 - 13a9z + 25a9z3 - 19a9z5 + 3a9z7 + a9z9 - 2a10z-2 + 14a10 - 32a10z2 + 34a10z4 - 21a10z6 + 6a10z8 + 2a11z-1 - 13a11z + 24a11z3 - 17a11z5 + 4a11z7 + a11z9 - a12z-2 + 6a12 - 11a12z2 + 11a12z4 - 6a12z6 + 3a12z8 - 2a13z5 + 3a13z7 + a14z2 - 3a14z4 + 3a14z6 - 2a15z3 + 2a15z5 + a16 - 2a16z2 + a16z4

Khovanov Homology:

trqj r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -5                     1 j = -7                   2 1 j = -9                 3     j = -11               3 2     j = -13             6 3       j = -15           4 4         j = -17         5 5           j = -19       3 5             j = -21     2 4               j = -23     3                 j = -25 1 2                   j = -27 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, 153]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 153]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[8, 17, 9, 18], > X[16, 7, 17, 8], X[20, 13, 15, 14], X[14, 15, 5, 16], X[12, 19, 13, 20], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {7, -5, 4, -3, 8, -6}, > {9, -1, 5, -4, 10, -2, 3, -8, 6, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-13 2 5 7 9 9 9 6 5 2 -3 q - --- + --- - --- + -- - -- + -- - -- + -- - -- + q 12 11 10 9 8 7 6 5 4 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 2 -36 3 -32 2 3 -26 5 -22 3 2 q + --- + q + --- + q + --- + --- + q + --- + q + --- + --- + 38 34 30 28 24 20 18 q q q q q q q 2 -12 -10 > --- - q + q 14 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 153]][a, z]
Out[8]=	8 10 12 6 8 10 12 a 2 a a 6 2 8 2 a + 6 a - 10 a + 3 a + -- - ----- + --- + 4 a z + 12 a z - 2 2 2 z z z 10 2 12 2 6 4 8 4 10 4 6 6 8 6 > 11 a z + a z + 4 a z + 9 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 153]][a, z]
Out[9]=	8 10 12 9 11 6 8 10 12 16 a 2 a a 2 a 2 a 9 -a + 7 a + 14 a + 6 a + a - -- - ----- - --- + ---- + ----- - 13 a z - 2 2 2 z z z z z 11 6 2 8 2 10 2 12 2 14 2 16 2 > 13 a z + 4 a z - 14 a z - 32 a z - 11 a z + a z - 2 a z + 7 3 9 3 11 3 15 3 6 4 8 4 > 3 a z + 25 a z + 24 a z - 2 a z - 4 a z + 15 a z + 10 4 12 4 14 4 16 4 7 5 9 5 > 34 a z + 11 a z - 3 a z + a z - 6 a z - 19 a z - 11 5 13 5 15 5 6 6 8 6 10 6 12 6 > 17 a z - 2 a z + 2 a z + a z - 11 a z - 21 a z - 6 a z + 14 6 7 7 9 7 11 7 13 7 8 8 10 8 > 3 a z + 2 a z + 3 a z + 4 a z + 3 a z + 3 a z + 6 a z + 12 8 9 9 11 9 > 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-7 -5 1 1 2 3 2 4 3 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 27 10 25 10 25 9 23 8 21 8 21 7 19 7 q t q t q t q t q t q t q t 5 5 5 4 4 6 3 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 19 6 17 6 17 5 15 5 15 4 13 4 13 3 11 3 q t q t q t q t q t q t q t q t 2 3 2 > ------ + ----- + ---- 11 2 9 2 7 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a153

L10a152 L10a154