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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-25/2 + 2q-23/2 - 6q-21/2 + 8q-19/2 - 12q-17/2 + 11q-15/2 - 13q-13/2 + 8q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-40 + 3q-38 + 3q-36 + 5q-34 + 9q-32 + 7q-30 + 11q-28 + 10q-26 + 8q-24 + 10q-22 + 4q-20 + 7q-18 + 2q-16 + 2q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- 2a5z3 - a5z5 - a7z-3 - 6a7z-1 - 13a7z - 11a7z3 - 3a7z5 + 3a9z-3 + 13a9z-1 + 17a9z + 6a9z3 - 3a11z-3 - 8a11z-1 - 4a11z + a13z-3 + a13z-1

Kauffman Polynomial:

2a5z3 - a5z5 + 5a6z4 - 3a6z6 + a7z-3 - 6a7z-1 + 13a7z - 20a7z3 + 17a7z5 - 6a7z7 - 3a8z-2 + 13a8 - 17a8z2 + 7a8z4 + 4a8z6 - 4a8z8 + 3a9z-3 - 14a9z-1 + 28a9z - 37a9z3 + 28a9z5 - 9a9z7 - a9z9 - 6a10z-2 + 24a10 - 33a10z2 + 14a10z4 + 5a10z6 - 6a10z8 + 3a11z-3 - 12a11z-1 + 21a11z - 20a11z3 + 15a11z5 - 6a11z7 - a11z9 - 3a12z-2 + 11a12 - 16a12z2 + 15a12z4 - 4a12z6 - 2a12z8 + a13z-3 - 3a13z-1 + 3a13z - 2a13z3 + 4a13z5 - 3a13z7 - a14 + 3a14z4 - 2a14z6 + a15z-1 - 3a15z + 3a15z3 - a15z5

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 = -4                     1 j = -6                   3 1 j = -8                 4     j = -10               4 3     j = -12             9 4       j = -14           5 7         j = -16         7 6           j = -18       4 8             j = -20     2 4               j = -22     4                 j = -24 1 2                   j = -26 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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 166]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 166]]
Out[4]=	PD[X[6, 1, 7, 2], X[2, 5, 3, 6], X[20, 13, 15, 14], X[10, 3, 11, 4], > X[4, 9, 1, 10], X[16, 7, 17, 8], X[8, 15, 5, 16], X[18, 11, 19, 12], > X[12, 19, 13, 20], X[14, 17, 9, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 8, -9, 3, -10}, > {7, -6, 10, -8, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 6 8 12 11 13 8 7 -q + ----- - ----- + ----- - ----- + ----- - ----- + ----- - ---- + 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 3 -(5/2) > ---- - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-40 3 3 5 9 7 11 10 8 10 4 7 2 q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + 38 36 34 32 30 28 26 24 22 20 18 16 q q q q q q q q q q q q 2 2 -8 > --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[10, Alternating, 166]][a, z]
Out[8]=	7 9 11 13 7 9 11 13 a 3 a 3 a a 6 a 13 a 8 a a 7 9 -(--) + ---- - ----- + --- - ---- + ----- - ----- + --- - 13 a z + 17 a z - 3 3 3 3 z z z z z z z z 11 5 3 7 3 9 3 5 5 7 5 > 4 a z - 2 a z - 11 a z + 6 a z - a z - 3 a z
In[9]:=	Kauffman[Link[10, Alternating, 166]][a, z]
Out[9]=	7 9 11 13 8 10 8 10 12 14 a 3 a 3 a a 3 a 6 a 13 a + 24 a + 11 a - a + -- + ---- + ----- + --- - ---- - ----- - 3 3 3 3 2 2 z z z z z z 12 7 9 11 13 15 3 a 6 a 14 a 12 a 3 a a 7 9 > ----- - ---- - ----- - ------ - ----- + --- + 13 a z + 28 a z + 2 z z z z z z 11 13 15 8 2 10 2 12 2 5 3 > 21 a z + 3 a z - 3 a z - 17 a z - 33 a z - 16 a z + 2 a z - 7 3 9 3 11 3 13 3 15 3 6 4 8 4 > 20 a z - 37 a z - 20 a z - 2 a z + 3 a z + 5 a z + 7 a z + 10 4 12 4 14 4 5 5 7 5 9 5 > 14 a z + 15 a z + 3 a z - a z + 17 a z + 28 a z + 11 5 13 5 15 5 6 6 8 6 10 6 12 6 > 15 a z + 4 a z - a z - 3 a z + 4 a z + 5 a z - 4 a z - 14 6 7 7 9 7 11 7 13 7 8 8 10 8 > 2 a z - 6 a z - 9 a z - 6 a z - 3 a z - 4 a z - 6 a z - 12 8 9 9 11 9 > 2 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 1 2 4 2 4 4 q + q + ------- + ------- + ------ + ------ + ------ + ------ + ------ + 26 10 24 10 24 9 22 8 20 8 20 7 18 7 q t q t q t q t q t q t q t 8 7 6 5 7 9 4 4 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 6 16 6 16 5 14 5 14 4 12 4 12 3 10 3 q t q t q t q t q t q t q t q t 3 4 3 > ------ + ----- + ---- 10 2 8 2 6 q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a166

L10a165 L10a167